The main advantage of twin spools over a single spool is the fact that the rotor speed better matches the airflow. On a large single spool engine the rotor speed is more or less optimal about halfway down the compressor section. So in effect this means it will be turning too fast for the first stages, and too slow at the later compressor stages. This can be partially resolved through blade design, variable inlet vanes, variable stator vanes and such but it will never be optimal. By creating more spools the N1 spool can use a lower speed (6 to 7000 rpm) compared to the N2 spool (over 13000 rpm) which better matches the local airflow through the engine. A triple spool engine is one step further of course and takes this principle to its next logical step. The drawbacks? More bearings, extra shafts, more complexity, and therefore better used on larger engines where the weight penalties will be least.
(N1 and N2 speeds as quoted are not based on any specific engine type!)