AFAIK GE's big twin spools weigh a lot more than RR's big trip-spool equivalents; that sounds like a bad thing for everyone except GE.
While Rolls has long touted the theoretical weight advantage of their three spool designs, they've had a much more difficult time making it work in practice. The RB211-524G/H engine for the 747-400 and 767-300 was actually significantly heavier than the corresponding two spool GE and Pratt engines (which played no small part in the flop of the Rolls powered 767).
All the engines have gotten much heavier as the fans have gotten bigger and the bypass ratios higher (the GEnx-2B fan is nearly a foot larger in diameter than the CF6-80C/E series engines, produces similar thrust, burns way less fuel, but weighs well over a thousand pounds more).
One of the problems with the 3 spool design is makes bearing design particularly difficult, and there have been a number of uncontained engine failures as a result (a bearing failure on an L1011 center engine tried to cut the airplane in two when the liberated fan came through the cabin, and we all remember what happened on the Singapore A380 a few years back
). As a result, all Rolls engines now have a number of elaborate, expensive, and
heavy safety systems to keep bearing failures from resulting in uncontained failures.
The end result is that, as far as I've been able to tell, Rolls engines on Boeing airplanes have not had a meaningful weight advantage relative to GE.