Time dilation in the movie Interstellar

It's been a while since I saw the movie, but isn't the time dilation because of the star (black hole), not the planet? If so, the same time dilation would have been in effect whether in orbit or on the surface of the planet.

A few minutes on the planet (meaning, deeper in the gravitational well of the black hole) corresponds to a couple decades where the ship is orbiting. Thus in the couple decades the ship is waiting, it only sees the one passage of the wave.

This is akin to the popular science depiction of an astronaut falling into a black hole. To an observer keeping a safe distance, the infalling astronaut appears to slow down. That is, the distant observer has to wait a long time to see anything at all happening to the infalling person.

Another way to look at this is to use a spacetime diagram. My extraordinarily rough but qualitatively correct attempt is below. Time flows up; the planet and black hole are somewhere off to the left. The left branch is the worldline of the crew that goes to the surface, while the right branch is the path taken by the ship staying in orbit. $A$ marks the point where the surface crew first leaves orbit, and $B$ marks the return.

Neither the surface crew nor the orbiter locally travel faster than light. This translates into their paths being more steep than $45^circ$ in the diagram. Light moves at $45^circ$ lines. As a result, any light signal either misses both paths or intersects them both exactly once. Even though traveling the left path you experience a few minutes between $A$ and $B$, whereas you experience years between the same points along the right path, you will see the same number of waves pass a given point on the planet over that time. Decades in orbit gets you the same amount of information as a few minutes on the planet.

I know that the science in movies is questionable and sometimes ridiculous but I would think this question would have been more obvious to the script writers.

Quite. Have a look at InterstellarWiki and note this: "Miller is a water world and the first planet in the system orbiting Gargantua. Miller takes its name from Dr. Miller, who landed on the planet and activated the "thumbs up" beacon".

When they visited Miller's planet they were almost killed by a re-occurring tidal wave.

Yes, and note this: "10 years before the Endurance crew travelled through the wormhole, NASA sent twelve landing pods through it, each carrying a scientist to assess a potentially habitable world. Miller was selected to land on this world. However, within a relative hour after her arrival, she encountered one of the massive tidal waves circling the planet. She was unable to negotiate the encounter and her landing pod was destroyed. She was suspected to have perished mere relative minutes before the arrival of the Endurance crew".

So, you find a planet, there's no land, but you give it the thumbs up inside an hour? It isn't just the science that's flaky in this movie. The plot is flaky too.

In a few short minutes they ended up spending 23 years there. When they were still orbiting the planet before going down wouldn't they have seen multiple tidal waves occurring one after another?

Yes. Because "Miller is a water world, covered in a seemingly endless, shallow ocean". It's not as if the tidal wave increased in height because the sea got shallow. They would have seen that 4000ft tidal wave. And they would have certainly detected it on radar. Regardless of time dilation.

Therefore why even bother landing on the planet?

Because it's a movie. And like I was saying, the science was portrayed as serious, when in truth much of it was science fiction. See this which backs up what Benito said:

"Being well within the tremendous gravitational field of Gargantua, time on the surface of Miller's planet passes very slowly relative to the rest of the universe".

If you were going to land on a planet, you'd orbit it and check it out first. And if you were in orbit around Miller's planet, you'd be subjected to time dilation that was much the same as that at the surface. The time dilation for GPS satellites is miniscule, it's 38 microseconds a day.

In addition to what should be a miniscule difference in time dilation between the water planet's orbit and the water planet's surface, I still don't understand the tiny difference in subjective time separating the two landers' arriving on the water planet's surface.

If two identical ships start a journey in the same frame of reference, and then travel identical paths that end up in the same frame of reference, then the subjective difference in time separating the two ships should remain constant.

For instance, in an automobile race two cars may go faster in some parts of the track and slower in others, but the difference in departure/arrival times should always remains the same. If Car B starts the race 5 minutes after Car A, then Car B should also finish the race 5 minutes after Car A, no matter how twisty the track. Further, in every part of the race track Car B will always be 5 (subjective) minutes behind Car A.

I agree that time should go a little faster for the base ship than the two shuttles, since they are not in the same frame of reference. (Assuming the base ship is orbiting the planet which I am not sure on.)