Correct, the main difference is what is sent trough the VPN. The browser based one obviously just covers the browser data, whereas the network based VPN should cover all network activity (depending on the VPN and its configuration.
Don't forget that with the browser plugin DNS requests may very likely leak, partially revealing your activity.
Other than that, the encryption itself and therefore the security of the tunnel may widely differ, depending on the type of VPN. If your browser "VPN" is for example the Opera VPN, it's more of a proxy than a VPN.
The third party, aware of your activity also changes. With the browser plugin, the providers of said plugin may have insight in your activity, whereas with the university VPN, they probably know what you're doing.
A "network based" VPN very much depends on the configuration, you can have a VPN active and connected and nothing is sent over the tunnel, because it's not configured that way. It's a question of routing and default gateways, as of what is sent over the tunnel.
There are quite a few differences and one could go into detail on every single attribute, but I think the vital point is, what do you want to protect and against whom. Starting from there, you can decide what makes more sense for you.
