Server and client certificates belong to unrelated certificate hierarchies. You can have your server cert signed by $whateverCA and still run your own mini-CA to issue client certs and validate client certs against.

TLS client certificates are strictly better than passwords because they don't provide an impersonator with a wildcard (and if you are running over the internet, especially with mobile devices, then you can get into that situation w/o being subject of a targeted attack). There are fairly few ways to achieve that other than client certs (SRP and SAE come to mind, both of which have virtually no deployment).