Pops to my mind, and painted with a broad brush, each point is actually more subtle...
Pros:
Security, we can lock down access.
Performance, less likely to have database connection problems, also latency should be reduced.
Agility, we can modify/create not only database information, but physical hardware/disk partitions and the like.
Growth room, if you start adding space/databases/whatever, it will likely be cheaper to scale a properly setup internal system than to upgrade your contract.
Cons:
Security, can offload external security to "the experts", however opens up inside job to those people.
Performance, assumbably the hosting company really knows how to tweak and run their servers for these databases.
Backups, any good external hosting plan is going to be in charge of your backup system (no trivial matter).