There were two big ones IIRC.
First, the 1812 model shortened the barrel from 44 inches to 42 inches. The change allowed the weapon to be more manueverable, and thus easier to use in close quarters. Second, the bayonet was moved from below the barrel to above, allowing it to act as a sight for the weapon, which increased accuracy.
The Model 1816 fixed other issues. The Model 1816 had a longer bayonet (for a total of 16 inches), a longer stock, and a shorter trigger guard. The last two were to make manufacture easier, faster, and cheaper.