The GE had the better fuel burn and reliability - especially early on - and the market share reflects that GE getting the lions share (IIRC, the CF6-80C2 powered more 747-400s than Pratt and Rolls combined).
Pratt upgraded the PW4000/94" with the ring case compressor configuration (mandated by AD due to takeoff surge issues with the original segmented case compressor configuration) which matched up very favorably with the CF6-80C2 installation. But by the time the RCC came on-line in the mid-naughts, the 747-400 was past its prime and was only being sold as a freighter.
The Rolls wasn't bad - fuel burn on a par with the pre-RCC Pratt - but it had issues with reliability, especially on the engine control side. Unlike the Pratt and GE, the RB211-524G/H wasn't a true FADEC - it was a "FAFC" - Full Authority Fuel Control. The vanes and bleeds retained the fluidic control from the earlier marks of the engine - and both the FAFC and fluid vane and bleed controls proved to be very problematic.