Alchemy didn't compile to ActionScript, it compiled to the underlying VM bytecode and it relied on special bytecodes added to the runtime specifically for Alchemy.
A more direct comparison in this case is NaCL - Alchemy-compiled outputs would not work on an old ActionScript VM, while one of asm.js's benefits is that the generated code will work on any javascript runtime, because it's just JS.
For the interest of historical accuracy, I must point out that Adobe Alchemy predates Emscripten by years.