The first fight is an ambush set up by Bane. He set the time, and the place. He knows things about Batman that Batman did not think he knew - that he is Bruce Wayne for example, off the bat (I would say that pun is unintentional, but I'd be lying). Batman is mentally thrown off. His tricks don't work:

Theatricality and deception are powerful agents to the uninitiated... but we are initiated, aren't we Bruce? Members of the League of Shadows!
....

Oh, you think darkness is your ally. But you merely adopted the dark; I was born in it, moulded by it. I didn't see the light until I was already a man, by then it was nothing to me but BLINDING!

There's also the discussion with Alfred that occurs earlier in the movie:

Bruce Wayne: If this man is everything that you say he is, then this city needs me.
Alfred: This city needs Bruce Wayne, your resources, your knowledge. It doesn't need your body, or your life. That time has passed.
Bruce Wayne: You're afraid that if I go back out there I'll fail.
Alfred: No. I'm afraid that you want to.

The Batman in the second fight turns up at Gotham, and Bane is completely blindsided - "I broke you". This time he's not thinking clearly, and as the fight goes on, Batman manages to trigger his rage. That punching the pillar scene isn't a loss of "skill"; it's a disoriented, extremely violent ball of rage punching with extreme strength as Batman is focused on staying out of the way of those fists. Bane's also distracted elsewhere - the vehicle with the nuke is the important part of his plan, and there's a strong sense that he means to be consumed by the nuke as well.

Batman has changed also - previously, even if he wasn't suicidal, his entire thing was about not fearing death. He has now learned a lesson about using his own fear to empower himself.

And so, the tables have turned.

In his first fight with Bane, 1) Batman subconsciously wants to die, and 2) Bane is more motivated.

Others have noted some of the tactical reasons why Batman lost the first fight and won the second. That's definitely part of it, but I'm going to argue for a more fundamental reason.

At the beginning of the film, Gotham is at a point where it apparently does not need Batman (the Dent Act led to the end of organized crime in that city). That is what Bruce wanted, since if he could stop being Batman he could pursue a relationship with the woman he loves, Rachel. With Rachel's death, he no longer has a reason to not be Batman, but Gotham also does not need Batman (at least it appears this way at the beginning of the film).

Given those things, Bruce no longer has a sense of purpose, which is evidenced by his reclusiveness and neglect of Wayne Enterprises.

Bane's emergence seems to give Bruce a reason to be Batman again, but he doesn't have an "exit plan," as his did before.

WAYNE: Rachel died knowing we'd decided to be together. That was my life beyond this cave and I can't just move on. She didn't. She couldn't.

Alfred also worries about this:

WAYNE: That's what you're afraid of - that if I go back out there I'll fail.

ALFRED: No. I'm afraid that you want to.

So, my theory is that Bruce subconsciously wants to die. I don't think that he wants Bane specifically to kill him, but he's okay with being killed as Batman in one way or another. Bane even notes this:

BANE: You don't fear death. You welcome it. Your punishment is to be more severe.

After Bruce loses to Bane and is in prison, he is annoyed by the fact that he wasn't killed. He asks Bane:

WAYNE: Why didn't you just kill me?

He even asks the other prisoners to kill him:

PRISONER: He asks if you would pay us to let you die. I told him you have nothing.

WAYNE: Do it for the pleasure.

At this point in the film he really wishes he would have just been killed in his fight with Bane.

So that explains the first part of the question.

As to the second part...

Batman wins the second fight because 1) he learns to fear death, 2) his desire to save Gotham as Batman is rekindled, 3) "falling" and "rising again" made him stronger.

In prison, the TV images of what's occurring in Gotham motivate Bruce to recover from his injuries and train. He's noticeably more driven than before. He sees Ra's al Ghul in a hallucination, who taunts him about his failure to save Gotham. This gives him additional motivation and purpose:

PRISONER: Why build yourself?

WAYNE: I'm not meant to die in here.

The film makes it clear that Bruce needs to learn to fear death. Batman can't make the "jump" without fear. With the safety rope, he doesn't have the appropriate fear of death. Without it, he does have the right amount of fear, which motivates him to make the jump.

WAYNE: I do fear death. I fear dying in here while my city burns with no one there to save it.

Presumably this fear of death, his anger over what Bane did to Gotham, and his training let Bruce beat Bane the second time around.

A big influence on the Nolan trilogy is The Dark Knight Returns and this battle looks highly influenced by Batman's fights with the mutant leader.

The mutant leader has similar physical attributes to Bane, he's younger, stronger, fitter, and has a driven savagery:

and indeed Batman comes off rather badly after their first fight:

Later Batman rematches with the mutant leader, having taken stock of the situation.

He realizes:

My mistake was to try to match his savagery

He had tried to fight the mutant leader (as he did with Bane) as a straight one-on-one, where Bane had the advantage of youth, strength and drive.

This time though Batman uses his experience, and his skills, as we can see in the question above neatly avoiding Bane's attacks, wearing him down.

Eventually having worn him down he can take him on straight, to a win this time

Of course in the Dark Knight Rises, Miranda intercedes, but you get the idea.

Your question:

Is the disparity between these two performances from Batman entirely the result of the fact that he broke Bane's mask in the second fight, or is there something else going on here?

has the answer:

Indeed, it is breaking the mask because Bane was vastly superior. Your video of the second fight is highly misleading because it is already near the end of the fight.

Bane: "So you came back to die within your city". Batman: "No. I came back to stop you".

Bane attacks with straight punch slightly from above.
Batman blocks the punch down with both hands and counterattacks with moving both arms up. No reaction from Bane.
Bane connects with a strong left hook.
Bane continues with a vicious right punch, Batman staggers.
Bane lands another punch with the left hand, Batman staggers back.

cut scene

Bane punches Batman strongly with the left hand.
Bane punches Batman in the midsection with an uppercut.
Bane punches Batman again in the midsection, Batman goes back.
Bane punches Batman's head with the left, Batman must roll with the punch.
Bane punches Batman's head with the right and really throw Batman back.
Bane takes a run and kicks Batman in the midsection, Batman stumbles on the stairs and goes up.
Batman awaits Bane and attacks with a strong left.
Bane blocks it and counters with a strong right into Batman's side.
Bane continues with a left swing to Batman's head and connects.
Bane punches with the right, Batman blocks with arms up.
Both are going into clinching, Batman tries to pinch Bane with the right arm and tries to hit him with the left.

Bane catches the left arm and both are going into a contest of strength which goes on for some time.

Until now Bane has hit Batman 11 times, Batman has only achieved one hit and it did not show any effect so far. Bane is vastly superior.

Now the key scene: Batman is able to break the fist lock by moving the arm down and finally connects with a surprise punch right into Bane's mask.

Bane is surprised and for a moment stunned.
Batman lands a second strong hit on Bane's mask with the left hand.
Batman uses his chance and connect with a third hit to Bane's mask.
Batman connects with the left elbow, Bane goes down. Batman holds Bane's head in a arm lock with the left arm.
Bane punches Batman with his head into the upper torso.
Batman goes back, Bane moves an idiot in his path out of the way.
Bane attacks with a weak left, Batman blocks and lands two terrific blows with the left and the right, the mask audibly snaps.
Bane holds a moment and the first time his look says: "Oh crap!"
Batman tries to go into the clinch, Bane throws him back and frantically tries to fix the mask.
Batman attacks and Bane hits him with an outward punch with the right, throwing Batman back.
Bane attacks Batman with the right, Batman blocks it.

Now Bane realizes that he needs to go to full attack and dishes out uncontrolled, vicious punches to put Batman down. This is the scene in the shown video.

It is now evident that the control and the agility of Bane had suffered greatly, he attacks 6 times and Batman easily dodges the blows and hits Bane again with two powerful blows on the mask.
Bane attacks and misses two times, Batman diverts the third blow, attacks with the right, Bane blocks, Batman puts Bane's defense down.
Again Batman hits Bane two times on his mask and kicks him into the hall.

Bane is defeated.

So the story is: yes, only the lucky hit on the mask put Bane on the defensive, before that he acted exactly like the first time: as unstoppable juggernaut.

Talia obviously knew Batman was going to meet Bane that night (their first fight): she and Bane were partners and in communication with each other, and an important part of their plan would be to eliminate Batman as a threat, so she would be aware of the trap Bane was setting. It's probably not coincidental that she chose that time to seduce Bruce; she was trying to deplete him of as much energy as she could so he'd be exhausted later when fighting Bane. (This is in addition to all the factors mentioned in the other answers.)

Another factor is simply because he feared the bat. When he saw the bat logo burned on the bridge, he was terrified of the man he dealt with.

I’ll give my view by sharing another Hollywood example. In his first fight with Bane, Batman had lost his fire, his “eye of the tiger” if you will. He appeared to have spent his years of solitude(following Harvey’s death) wallowing in self pity and depression, welcoming death and allowing his mind and body to deteriorate to a state reminiscent of Howard Hughes(which is referenced by peeing in “mason jars” “When the man approaches with the bag of cookies I want him to hand them to me with his left hand...” Soon after his reemergence Batman battles his toughest foe to date and understandably loses.

Which brings me back to Rocky 3. Rocky became complacent and did not heed Micky’s warnings of how dangerous Clubber is. He didn’t have the “eye of the tiger” and loses spectacularly. He and Batman undergo similar character redemptions complete with training montages and they both come back and defeat their greatest challenge after suffering strife and growing from their failure, and they both become stronger after their journeys.

Batman was the hottest fighting commodity when he was properly involved in his aim of destroying Joker and thugs in the Dark Knight but when he appeared in the Dark Knight Rises, he got confronted with the doctor saying to him that: "You are Physically wrecked MR Wayne".

Bane encountered with the old and out of practice Batman that is why he remained dominant. When Alfred warned him that Bane could deliver you a surprise damage, Bruce showed a non serious reply to his warnings and at the first dual he discovered that why Alfred was worrying about his feud with Bane.

In the pit, when Batman met with the vision of his biggest dysfunction, Du Card RAS AL GHUL. He got his self triggered again and somehow not so comprehensively, betrayed bane by hitting couple of punches on his mask. If Bane gets an encounter with the well trained and ferocious Batman as he was in The Dark Knight, Bane could be troubled.

So the sum up is, Chris Nolan portrayed that even by facing decline physically, mentally and economically, a man can rise and grip the things up.

So, for your question, there are many answers, but I will provide you the main reasoning for why Batman gets exterminated by bane in the first fight, and wins the second.

Why Batman lost his first fight:

As seen at the start of the movie, Bruce Wayne is holding a stick and walking helplessly, depressed, with anxiety because he was accused for the cause of death of Harvey Dent, 8 years ago in "The Dark Knight". Now he decides to take on his role as the caped crusader once again after Bane poses a serious threat to the city, with starting a "fire" as mentioned briefly while he was hijacking the plane. It became clear that Bane's intentions for Gotham were terroristic to the city, especially with the nuclear bomb near the climax of the movie. However, let's talk about Batman's first encounter with Bane, and why that went wrong. The city was in need of a savior, and Batman noticed that, and decides to find Bane and end this, despite being warned of Bane's skills and tactics by Alfred. Now he was not ambushed but surprised in the fact that Cat Woman turned on him, as she did not want to threaten her own life. Batman hears Bane interrupting his talk with Catwoman and turns to face him on a bridge like structure, with Bane right across. Now before this fight even started there was already a major flaw that Batman had not realized with fighting Bane, and that was his overconfidence. He only saw Bane as a mercenary under the league of shadows and R'as al ghul. He thought that since he defeated Ra's Al ghul himself, there was no way he would lose his fight with Bane. He underestimated Bane's power and right as the fight had begun, you could tell that this would not end well for the dark knight:

I mean, just look at how slow and weak his punches at the start were, Bane wasn't doing anything at the beginning, he wasn't even trying. Batman literally starts off with holding Bane's vest and it takes him a whole 2 second from him holding his vest to his first punch. After he uppercuts and swings a right at bane, and as Bane catches his hand, he reveals what Batman is really missing. This was just an exhausted and weak dark knight, I mean he is supposed to know all 127 martial arts, and as seen he is just making a fool of himself. And this relates to the second reason for why he lost, it had been 8 whole years since he has taken the role as Batman, he is weak and merely lost his skill. He was not at all ready to fight Bane. Bane even says: " Peace has cost you your strength, victory has defeated you " he is highlighting how his exhaustion and inexperience lead to a loss of strength and power. He didn't listen to Alfred, and that ended up costing him his back. He invited death in the hands of bane, despite knowing the city needed him. This was an unmotivated, weak, powerless, unskilled, overconfident, broken dark knight. I mean he lost so much skill, that later in the same fight, he was grabbing Bane with his right and was punching with his left continuously while deprived of strength and energy.

Now let's talk about why he won his second fight with Bane:

After losing his fight, he gained his motivation by prisoners and watching Gotham on a TV, he knew they needed a hero. He worked out, trained, rose back up again. R'as al ghul, came in his dream to wake him up, and remind him of his defeat, which motivated him even more. After successfully making the jump with fear, he overcame his biggest weakness. He grew stronger, and was ready for Bane. He fought with more skill and power and more intelligence, as he figured Bane's weakness and abused it to his advantage. Batman was not weak and was in better shape and grew stronger after the climb, he fought his fears with intense motivation as the caped crusader.