I used to work ar Royce's in the Noise office, so have a bit of perspective on this.
The noise requirements on all modern aircraft are very demanding, and there isn't a lot to choose between the direct competitors.
RR managed to get a better result than Pratt's on the B757, that meant it could operate less restrictively at places like Washington National and Orange County (John Wayne Airport), that have tight noise regs.
Being mostly 3-shaft engines, RR have a slightly different characteristic. The fans are slightly slower and the IP compressors slightly faster shaft speeds, as the separate shafts allow better matching and straighter ducts.
The rules used to allow a bit of "trade-off" between noise levels at take-off and approach. This facility was used differently by the different companies at times, so one type that's quieter on approach may be noisier at take-off.
Its been a while now since I was involved. Hope this helps.