Triple spool means that the engine is designed with three different stages of compressor blades, intake air is compressed in three successive steps before mixing with fuel in the combustion chamber. Yes, it is true that RR-RB211 engines are noisy compared to PW-2037 engines, they have a lower bypass ratio but they deliver more thrust. RR engines, in the case of the Boeing 757 which is the aircraft related to the scope of this topic, deliver 43000 lbs of thrust each, while PW engines deliver only 37000 or 38000 lbs of thrust. I have flown on both RR and PW powered 757's and I do feel there is such a difference. On RR powered 757's, during climb at full power, you almost feel like you are travelling on a rocket!
American, British Airways, Continental and US Airways operate the 757's with RR engines. Eastern had RR engines on 757's too. Delta, Northwest and United chose instead the PW engine on 757's. That doesn't surprise me because Northwest and United always had a special relationship with Pratt & Whitney. TWA's 757's had PW engines, that's why you'll see sometimes American 757's with PW engines. It's easy to spot a PW engine on a 757. The PW engine has a larger cowling surrounding the largest fan to increase the bypass ratio, the RR engine dosn't have that. The cowling on the RR engine surrounds the whole machinery, there is a nozzle but you don't see it. That's how I can tell an American 757 comes from TWA.