Dinger,
They do have different engines in 319, 320 and 321.
For example Austrian uses the following models:
A319:
CFM 56-5B6/B - 23,500 lbs
4,500 km
A320:
CFM 56-5B4/2P - 27,000 lbs
4,300 km
A321:
CFM 56-5B1/2P - 32,000 lbs
-111: 2,360 km
-211: 3,500 km
Also B737-600/700/800/900 uses different engines, more powerful on the bigger models.
Actually might use almost the same engines as the airbus A32S's..
etc.
EDIT: Added the range with full payload