Why?

Supes' one weakness other than Kryptonite is attack from another dimension. (Mister Mxyzptlk proved this.)

The TARDIS hands The Doctor the win for the fight, even if no one opted for trying to actually destroy the other.

Unite against a common foe, specifically the one making them fight because they've got a vested interest in them being bogged down for a bit while trying and incapacitate each other. It's the sort of thing that fits both their characters.

That it results in both Superman and the Doctor coming after fanboyish internet lardovores who derive vicarious satisfaction from "their guy" beating up someone else's in what amounts to a fictional character cockfighting ring is merely a side benefit. =]

I don't know much about DW, but which Doctor, exactly? And why would they fight? I think it would either end with Superman smashing the TARDIS, or The Doctor traveling through time to get a piece of kryptonite. Otherwise, they would probably be at it until one of them either dies of old age, or passes into the public domain

The Doctor can't beat classic Supes, short of killing him, which he wouldn't do, and while Superman can physically overpower the Doctor, he wouldn't because there's no reason to beat up an unassuming British fellow in a coat.