What do you mean by using Tor and a VPN service? Do you mean accessing Tor through the VPN? Or accessing the VPN through Tor? It seems like using Tor is a waste of time if you then go ahead and authenticate yourself somewhere. Or is this just in the context of trying to break out of a restricted environment? (rather than looking for anonymity)
Good question, I have been wondering the same for a while.
I have read somewhere the point is that one is anonymity provider, other - encryption to hide from your ISP.
If you think about it, you'd be using Tor to connect to either public or roll your own VPN. The issue with using Tor is that all exit nodes are (probably) monitored. Some websites even refuse to serve you, as they have a list of exit nodes. Use of public VPNs is easier, as semi-officially they track you, hence they can be trusted by websites. Your best option is use Tor & custom VPN with a clean IP address. Trouble here is leaving payment data and you'd still probably would trigger some alarms by connecting from a registered Tor exit node...
I meant using a VPN through TOR, between the client and the VPN service. This prevents the TOR chain from sniffing, at the cost of the VPN service potentially seeing your client.