Another advantage for the 3-spool that I don't think I have seen here yet:
3 spools allows the engine to be smaller in cross-section because there is an extra spool developing some of the power. This reduces some of the overall drag and could also be a positive where, for example, an aircraft design requires more clearance from the ground when 'hanging an engine under the wing. On the BA 777s a RR Trent 800 has 2 feet less diameter on the engine casing than a GE 90.