Oh I hate to say this. Actually, Amsoil was first. In fact, their tag line is "The first in synthetics" By a month or so. It was a marketing race. Mobil introduced their 5w30 shortly after Amsoil introduced their 10w40. And oh, did that Mobil 1 leak out of those engines of that era!
I worked in a Mobil station at that time, and my friend was an Amsoil dealer.
Amsoil was originally spelled Amzoil, but there was some trademark infringment, so they changed the spelling.
To that end, they are both great oils, and you can't go wrong using them.