A submachine gun is an automatic weapon that uses a pistol cartridge. It's generally between the two in weight and size. The idea is to have a weapon that has a superior rate of fire and power compared to a pistol but isn't as bulky as an assault rifle.
A sub-machine gun or SMG is a weapon that can fire fully automatic, but fires ammunition that is smaller than typical rifle calibers. . 45 ACP or 9mm are typical.
I want to point out that the Army does not name any of its weapons a 'sub-machine gun. ' The term itself refers to a rifle that fires in 3-round bursts.