3-spool, advantages...
Fuel efficiency.
Quiet, more so generally than a 2-spool design.
Disadvantages...
More expensive (generally).
More complicated, and heavier.
The first 3-spool engine (RB.211) was used on the Lockheed TriStar, exclusively, and after a short problem start, provided superior performance...especially in the fuel consumption and noise departments.
Three spool Rollers are
very good engines...the best in the business, in my humble opinion.
The Dart wasn't bad, either, although far outdated now, it also was superbly reliable.
Bar none.
If I was designing an aeroplane now, today, it would have Rollers, and nothing else.
The Brits done it right.