In fact there were four different kinds of this automatic carbine and five different names for it: Mkb2 H(Haenel), MP 43, MP43/1, MP 44 and Stgw 44, the last three are identic weapons.
I hope I get the english technical terms right:
The MP43 differs from the MP43/1, MP 44 and Stgw 44 in screw thread at the barrel muzzle and in the shield at the front sight. The MP 43 also used the standard grenade launcher from the 98k while MP43/1 used the ad hoc developed Gewehrgranatgerät Typ 43.
There are no changes between MP43/1, MP 44 and Stgw 44 i´m aware of.
And the change from MP 44(Maschinenpistole/machine gun) to Stgw 44 (Sturmgewehr/assault rifle) is due propaganda purpose. Not MP 43 to MP 44! I guess it sounds more intimidating