The advantages boil down to redundancy and capacity.
If your typical PC is a pickup truck, able to carry a small load of bricks, then you could think of the mainframe as a WHOLE FLEET of commercial dump trucks, each able to carry many pickup-loads of bricks. The redundancy is built in to maximize uptime and reliability. If you need to move huge quantities of bricks, it doesn't matter much if one or two dump trucks (in your large fleet of dump trucks) break, or go down for maintenance...the bricks keep moving.
Essentially, mainframes can process MASSIVE amounts of information (compared to mini computers or micro computers / PCs), and do it very reliably.
The disadvantage of mainframes? Cost of hardware, special operating systems/software (more cost there, too). You typically won't see a mainframe used unless some organization with really deep pockets has a mission-critical function that requires massive amounts of information to be processed, in a very reliable manner.