What are the advantages of 3 spool over 2 spool turbofan engines?
There are two advantages;


It's easier to match the compressor rotational speed with the turbine. Typically, the first stages of the compressor want to rotate slower than the turbine is required to rotate. The turbine needs to spin faster to help obtain as much energy from the airstream as possible, but initial stages of compression prefer to spin slower. With 3 spools instead of 2, less compromise is required.
A greater pressure rise per stage can be achieved. This occurs because each compressor in a 3 spool engine (fan, LPC, HPC) will have fewer stages than each compressor in a 2 spool engine (for the same overall pressure ratio). A 5 stage compressor requires less compromise than an 8 stage compressor. The front stages prefer a high airflow to avoid stall, but the back stages can't always swallow the same mass flow, and can cause a backpressure which can lead to stall. Hence, air flow is often bled off in the middle of the compressor, to avoid stall. This is air that has been partly compressed, absorbing work, but isn't available to generate thrust. Hence, it's a parasitic load. So, the compressors in a 3 spool engine are better optimised than in a 2 spool engine, because this situation isn't as severe. Hence, the designer can be a little more aggressive in the pressure rise per stage the compressor achieves, because the compressor is less prone to stall. Thus, a 3 spool engine, in theory, can achieve the same overall pressure ratio, as a 2 spool, with fewer stages. This means, a lighter weight engine.