The advantage of a SAN
It's simple: the advantage of a SAN is that it decouples your storage space from your servers.
This means that you can deal with each side independently; you can add storage or move storage without having to change the server hardware, and you can replace (and add) servers without having to do anything to your disks. In turn this means that you can use common garden variety servers instead of big monster servers that are designed to hold loads of disks, which means it is much easier to maintain and replace them.
(You may also be able to use relatively commodity disk enclosures, instead of monster huge specialized ones, although this depends on your SAN.)
This decoupling also simplifies failover and recovering from hardware problems in general, because it means that a server failing doesn't automatically take down some disk space and vice versa. (Merely using a SAN does not necessarily make your environment more reliable, though.)