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?
The gameplay never changes.
Videos like this one are fake:
As Strix says in his comments, the probability of each pipe is considered independently. Take a coin toss as an example:
Suppose that we have just tossed four heads in a row, so that if the next coin toss were also to come up heads, it would complete a run of five successive heads. Since the probability of a run of five successive heads is only 1⁄32 (one in thirty-two), a person subject to the gambler's fallacy might believe that this next flip was less likely to be heads than to be tails. However, this is not correct, and is a manifestation of the gambler's fallacy; the event of 5 heads in a row and the event of "first 4 heads, then a tails" are equally likely, each having probability 1⁄32. Given that the first four rolls turn up heads, the probability that the next toss is a head is in fact, 1/2.
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.