Pilots who are rated on one type can fly it, regardless of what kind of engine is attached. I do not know of any exceptions to this.

In terms of why an airline may choose different powerplants, one reason is fleet commonality. BA had 747 engines bolted to their 767s (some of the few such examples in the world) because they have so many 747s that it significantly reduced maintenance costs. Another option might be that a particular engine manufacturer might give discounts to the airline as well. It's a pricing game, of course.