Wikipedia mentions that Wasm is faster to parse than asm.js, and I'm guessing Wasm might be smaller, but is there any other reason? I don't think there's any reason for asm.js to have resulted in slower execution than Wasm.

Both undersell and oversell. There are still cases where vanilla JS will be faster.

And AFAIK asm.js is the precursor to WASM, like the early implementations just built on top of asm.js's primitives.