What does the distance to the Pokémon change?

Yes, distance to a Pokemon very much affects how likely it is to catch. It is not true, however, that closer Pokemon necessarily have a higher catch rate than further away Pokemon, but as Pokemon get further from you their multi-colored "target ring" gets smaller.

From the question What does the size of the multicoloured circle mean?:

The size of the ring represents your increased chance of catching the pokémon.

You have the greatest chance of capturing the Pokémon while the colored ring is at its smallest diameter. At the opportune moment, fling the Poké Ball toward the Pokémon.

(source)

When a Pokemon is closer, the target ring at its smallest is bigger than the same respective size target ring of a further away Pokemon. For example, if a Pokemon is close by, and the target ring minimum is 1cm in diameter, then further away, the target ring minimum could be 0.25cm. This means that it is easier to maximize capture rate (like getting an "Excellent!" throw, when the target ring is the smallest) when the Pokemon is closer.

This doesn't mean that you can't achieve the same capture rate when the Pokemon is further away, but it is just harder to achieve.

EDIT: This is probably well known at this point, but when a Pokemon escapes from a PokeBall, it usually bounces back further away than where it was originally, making it harder to catch.