Check with Event Viewer, but I guess you are having the same problem as myself and a number of others.
It's the fact that Windows 10 doesn't come with SECDRV.SYS driver, and the one that comes with the game is a 32-bit driver that isn't signed and won't work. This driver is part of the copy protection mechanism on AOM and other games.
The driver was provided with Win 8.1 64 bit, which is why it worked on that platform.
Which all together means that AOM (and other games that use the same copy protection technology) can't work on Windows 10 until Microsoft provide a compatible driver.
To the best of my knowledge, there is as yet no statement as to whether it will ever be supported ...