birch is more expensive and a bit harder to work with. mdf is so much easier to cut/work with/ fix mistakes. It depends on what your skill level is and what tools you have at your disposal. If you have a table saw then hell yeah go with birch. But if you're like me and only have a circular saw and a steady hand mdf is much easier to work with.
In terms of weight the cheaper 'birch' is lighter for sure but real birch is about the same weight as mdf but can take a lot more abuse. The birch most people use is a birch veneer then one ply of pine, a ply of birch, back to pine until it is 7 ply. I got some baltic birch 13 ply wood for $70 and that is real birch and some nice stuff. $70/sheet though. I'd stay with mdf - $25/sheet around here.