David Charlesworth uses sycamore in his planing demonstrations (I believe) because it is such an easy wood to work with. It's difficult to think of a better behaved wood for chiselling than sycamore. I don't know how easy it is to work birch in comparison, but it's unlikely to be any easier.
Would it be possible for you to take a photo of your work and post it so that we can help you out? It's much easier to offer advice when we can see what the problem is.
Not all that easy to answer really.
As a woodturner, I would consider that birch is softer than sycamore, but the sycamore would work better.
The trouble with birch is that it tends to start rotting before you've even finished chainsawing the tree down :-k