Why do men and women have different sporting capabilities, e.g. why is the women's hour record significantly shorter than the men's?

For a given level of fitness, wattage is proportionate to weight: that is, a 50-kg rider who can generate 4 W/kg could be considered to be at the same level of fitness as a 75-kg rider who can also generate 4 W/kg.

When riding on level terrain, though, your power/weight ratio doesn't determine your speed, it's your raw power output. So the 75-kg rider (generating 300 W) will beat the 50-kg rider (generating 200 W) every time. On climbs, the smaller rider will have the advantage because gravity becomes an important factor.

The average woman is smaller than the average man, so she'll be at a relative disadvantage in the hour. Of course, average people don't attempt the hour record, but until a woman who's about as big as a man attempts it, we probably won't see women overtake men in that event.

It's worth pointing out that there are some cycling records held by women: fastest motorpaced speed on a bicycle (Denise Mueller-Koronek), and highest annual mileage bicycled (Amanda Coker). Women have won both the Transcontinental Race (Fiona Koblinger) and the Trans-America Bike Race (Leal Wilcox).

You pointed it out yourself: “Men are born with a biological physical advantage.” It’s mostly due to a naturally higher level of testosterone.

If you compare men and women in various sport world records you’ll find roughly a 10–20% advantage for men. Usually a bit less in endurance sports and older age groups.

Of course individuals differ (it’s a bit like a Gaussian distribution) and there could be a woman who’s so talented and exceptional in every aspect that she can beat men at the highest level. But that’s highly unlikely, especially since most racing authorities have different limits on testosterone (even if no doping can be proven) for men and women. A woman with a naturally high testosterone level wouldn’t be allowed to compete and it’s doubtful whether any amount of talent can make up for that.

An example which comes to mind is Chrissie Wellington, who managed to be among the top in overall (men+women) score in world class Ironman Triathlon events several times:

In July [2011], Wellington bettered her own ironman-distance world record at Challenge Roth by exactly one minute, to 8:18:13. Her marathon time of 2:44:35 was also a new world record. Only four men finished in front of her, and only one man, the winner Andreas Raelert, who also set a new world record, was able to beat her marathon time.

(emphasis mine)

Of course all of this is about world class level. On an amateur level (up to about federal championship level) it’s totally possible for women to overcome that 10% disadvantage with better training and beat men (who train worse, are less talented etc.).

Yes, Males have a significant advantage over females, on average, for the kinds of activities society turns into sports. Males are typically larger, taller, have greater lung capacity, larger hearts, and larger oxygen carrying capacity. Male and female bodies have different ratios of muscle and fat even for two average adults of the same weight, with males having more muscle.

But, that does not mean that a woman cannot reach the potential of a man, right?

So, let's look at this statement in depth, and what man and what women we're comparing against. What I described above, is a comparison between two average people. The average person does not exist. And, the difference between the slowest male and the fastest male is significantly larger than the difference between the average male and the average female. So there are going to be tons of women faster than men.

So when you are competing in competitions that are full of amateurs with day-jobs that are in it for the fun and challenge, women can absolutely take the podium. To win, you need to persevere and work hard.

However, when you are looking at the world records, or best-in-the-world class competitions like the Tour-de-France, that variance becomes irrelevant, because we're not looking at average males or females anymore. There, you are competing within a segment of the population representing the very fastest, most capable specimens our species can produce. To compete at these levels, you don't get there by just working hard and persevering. Everyone there is doing that. You also need the genetic advantage to get you that extra edge. And unfortunately, in that kind of competition, women are at an insurmountable disadvantage.

Sexual dimorphism is a real thing, and humans are no exception to it.

Note: There are some events where for whatever reason, women do seem to have an advantage over men, such as in ultra-distance swimming.

One answer specific to UCI events such as the hour record is that different allowable biological parameters between genders put women to be at a disadvantage. Specifically, women must have a hematocrit less than or equal to 47 (actually 48 since there is a 1% allowable margin), while men must be less than or equal to 50 (+1% margin). Since both men and women can achieve hematocrits well above 50 with the help of transfusions or EPO, it is common for elite athletes to compete with hematocrits as close to the allowable limit as possible. By setting a lower limit for women than for men, the UCI makes it harder for women to reach the same level of performance as men.

Similarly, the UCI requires female athletes to have testosterone levels below 10 nmol/L, while men can have much higher levels (as long as there is no evidence of synthetic testosterone). Since anabolic steroids such as testosterone are helpful in building muscle mass and improving recovery after training, the cap on testosterone levels for women puts them at a disadvantage relative to male athletes.