Though you're technichly not supposed to play LAN together on computers all using the same CD Key, it won't cause a version mismatch. The only possibility is that they're not patched alike -- you'll have to do a full reinstall on both versions, and make sure after you've uninstalled that you delete the Age of Empires III directory before installing again -- it could have some extra files in it that are messing up your game. That directory could be in one of a few places, based on how you got the game:
Retail or GFWL: C:\Program Files\Microsoft Games\Age of Empires III\
Steam: C:\Program Files\Steam\steamapps\common\bin\Age of Empires III\
Impulse: C:\Program Files\Microsoft Game Studios\Age of Empires 3 - Complete Collection\
Change "Program Files" to "Program Files (x86)" if you're running a 64-bit OS.