Is there any variation or end to Flappy Bird?

No

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.

Source: en.wikipedia.org/wiki/Gambler's_fallacy

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.