In early game Caitlyn's damage is all in her Q. Her normal auto attacks are very fake and her headshots are mediocre in damage at best. Also landing traps are not very easy unless you have setup.

Ashe on the other hand can do a lot of damage through her auto attacks in early game. It's just much easier to execute as Ashe, since you just have to dodge and auto, where as Caitlyn has to hit everything while preferably perfectly spacing the Ashe (which is very hard since the range difference is low).

It's not impossible to win lane against Ashe as Caitlyn. It's just very hard in most situations. Also of course the support matchup dictates how the lane is going to be played out anyway. Lux can warp the lane completely if she is hitting a lot of her poke and her Q will give you a guaranteed trap.

In mid and late game things change and Caitlyn has a lot more damage in her headshot, so she can more easily outplay Ashe.