Birch plywood (often called Baltic Birch) is used for applications where very high quality and stability is desired. The big difference between birch plywood and regular construction grades of plywood is in the quality of the wood used and (most importantly) the number of laminations. It's the combination of many laminations and quality wood without any defects or voids that gives birch ply its superior quality. One place where this comes into play is in the dovetail or finger joints. The outer laminations of regular plywood may flake or break off, but the birch plywood gives perfect results. For a given size piece I think the birch ply would be denser, stronger, and more resistant to twisting or distorting, but I do not have any references to quote on this. It is just my experience of many years of woodworking.
I rooted around in the shop and found some 1/2" birch plywood. Here's a photo of it with a piece of regular 3/4" plywood on top. The difference is readily apparent.
Of course, both types of plywood will be found in various grades of quality. In my experience, even the cheapest birch plywood is better than any regular construction grade material. I am not certain just how these properties would affect the sound of a cabinet, but when I built a 2 X 12 cabinet with Celestion Vintage 30's in it I chose to use good Baltic birch I purchased at a lumber yard about 45 miles from home.