Is there any variation or end to the game? At some point, at some score, does the difficulty increase, does the environment change, is there ever an end to it?
As Strix says in his comments, the probability of each pipe is considered independently. Take a coin toss as an example:
The game only becomes increasingly difficult due to player fatigue etc.
No, the game does not increase game-wise.
It does, however, get more difficult. This is not because the game itself is getting more difficult, but because the longer you play, the higher the probability that you will make a mistake.
This is called difficulty through duration.