I have both.
Must admit, half chaps are just so much more comfy than boots. And when dirty you can just stick them in the wash. I do wear tall/long boots for competitions though.
We ended up having to break down and get my daughter tall boots. She needed them anyway for showing, but she wears them schooling too because her half chaps starting rubbing blisters on her legs. I guess they were too loose and they caused the seams of her jeans and breeches to rub her legs. The boots protect her legs better. But half chaps are a much cheaper way to go, if you can get away with it.