Assume there's no scaling penalty for storing all the clients in one database; for most people, and well configured databases/queries, this will be fairly true these days. If you're not one of these people, well, then the benefit of a single database is obvious.
In this situation, benefits come from the encapsulation of each client. From the code perspective, each client exists in isolation - there is no possible situation in which a database update might overwrite, corrupt, retrieve or alter data belonging to another client. This also simplifies the model, as you don't need to ever consider the fact that records might belong to another client.
You also get benefits of separability - it's trivial to pull out the data associated with a given client ,and move them to a different server. Or restore a backup of that client when the call up to say "We've deleted some key data!", using the builtin database mechanisms.
You get easy and free server mobility - if you outscale one database server, you can just host new clients on another server. If they were all in one database, you'd need to either get beefier hardware, or run the database over multiple machines.
You get easy versioning - if one client wants to stay on software version 1.0, and another wants 2.0, where 1.0 and 2.0 use different database schemas, there's no problem - you can migrate one without having to pull them out of one database.
I can think of a few dozen more, I guess. But all in all, the key concept is "simplicity". The product manages one client, and thus one database. There is never any complexity from the "But the database also contains other clients" issue. It fits the mental model of the user, where they exist alone. Advantages like being able to doing easy reporting on all clients at once, are minimal - how often do you want a report on the whole world, rather than just one client?
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