Birch, when left "white" is about the same color as maple is. It is almost utterly devoid of any color variation in the wood. The grain is as coarse as most walnut. When it is stained (if it is stained well, which it rarely is) it can look somewhat like incredibly plain, light colored walnut.
Beech can have a tad more yellowish color in the wood, but it is usually stained with a horrible looking pigment stain. Beech is MUCH tighter grained than birch, and the two woods look nothing alike. Beech will have rays in it. Hard flecks that radiate out from the center of the tree. It can be quite gaudy (though usually not nearly as much as Sycamore can be!). European red beech is often used on European guns, and if the wood is left clean, it has a bit of a pinkish color to it (by the way, if left in the sun, like in a window, it will turn VERY yellow), but otherwise is essentially indistinguishable from American beech. When stained, you'll never be able to tell the difference. Beech is always VERY hard, and also hard to work, as it is rather splintery. Birch can be varying degrees of hardness, but I haven't seen any that was really "soft" yet.
If you have rays in the wood, it is beech. If not, more than likely it is birch. "select hardwood" means absolutely nothing. It is used to make the wood sound like it is as good as walnut! :winking: