For hunter/jumper we pull manes. Cutting makes all the hairs exactly the same length and 1) I don't think it looks that great. 2) It's hard to do if the horse has a thick mane and, 3) when all the hairs are the same length it makes it a lot harder to braid for shows. (hunter braids). I know that a lot of the Jumpers get their manes cut because they stay longer and don't have to be braided.
The essential joy of being with horses is that it brings us in contact with the rare elements of grace, beauty, spirit, and fire. ~Sharon Ralls Lemon