The differences in plywood usually revolve around the number of plies (a.k.a. layers) and the quality of the veneer.
The best birch plywood, which you probably won't be able to find at a big box store, is Baltic Birch. It has twice as many plies as "normal" plywood and is guaranteed to be void free. Finnish Birch is the same as Baltic Birch, but the glue used to bind the plies together is waterproof (or water-resistant). You an easily tell the difference between the two by looking at the edges: the waterproof glue used in Finnish Birch plywood leaves darker lines between the plies than the glue used in Baltic Birch. Because of the increased number of plies, Baltic Birch routes very well. You could, for example, cut a round-over at the edge and have it come out looking perfect. Try the same thing with regular plywood and you'll end up with something that looks like it was done by a drunk beaver!
In your case, the thing to do is probably try to find out the "grade" or "classification" of the plywood. This will probably be something like "A-1", "A-4", or "C-D". There's a lot of info on the internet about
grading. Also keep in mind that the grades aren't necessarily uniform. I've purchased A-4 cherry plywood sheets that looked every bit as good as the A-1 sheets that cost 50% more! Finally, there may also be a difference in how the veneer was sliced from the log. I believe that "plain sliced" tends to look more like real wood and costs more money than "rotary sliced" veneer. Or I may have it exactly opposite; you'll be able to tell by the price, though! And perhaps by looking at it: one method has grain patterns that are more attractive than the other.