This seems to be about multi tenancy - there are several SO answers on this topic.
It's a multi-dimensional trade-off.
If you have "hundreds or thousands" of clients, those clients presumably have related entities - orders, contact details, etc. So you're almost certainly not just considering have many "client" tables, but essentially many (logically and/or physically) separate databases with the full schema to support your business domain.
That's a big decision - you now have to figure out how to deploy thousands of instances of your software, support thousands of environments (even if they are logical, rather than physical), make sure each of them is configured properly etc. The good news is that this may make security and provisioning easier, and it should allow you to partition clients to ensure high performance.
That's a big decision - I'd say it's an order of magnitude harder than building an app with a single database. I'd really only want to make that decision if I were 100% certain it's necessary. Modern databases can easily query across very large datasets as long as you can use indexes; hundreds of millions of rows are not particularly scary.