MOST RR engines are 3-spool (but not all).
I'm no expert on this, but the discussions I see say that the 3-spools are more efficient in climb and over short ranges, and the 2-spools are more efficient over long cruises at high altitudes.
From the airline PoV, one or the other might be more desirable depending on route structure (domestic vs. transoceanic).
From the pilot PoV (as jetjock says) you'd have an extra N gauge per engine to monitor.
RR Trents rotate clockwise (as seen from the front) and GEs rotate counterclockwise. At least the fans on the front. That may be related to the third spool and efficient energy recovery with three turbines - or it may just be the old "British drive on the left, Yanks drive on the right" habit kind of thing.