In essence, VPNs depend on proxies, but proxies do not in and of themselves make up a VPN.
Proxy server is a general term and can mean all sorts of things, including reverse proxy. A proxy isn't necessarily private and isn't necessarily encrypted
A proxy can
- let you be anonymous by hiding your IP from the destination server
- let you bypass network restrictions by hiding the source server IP from your parents' router controls
- work as a traffic cop to direct requests to a set of servers (i.e. reverse proxy).
Basically a proxy is any server that sits between you and the destination server that makes requests on your behalf and/or serves requests on behalf of the server, unlike a network switch that is just relaying your requests.
It's maybe like the difference between having your home phone set to forward your calls, and having an answering service that picks up the phone, receives the message, then hangs up and calls you and gives you the message.
So in that example, the forwarding is just a switch. The answer service is a proxy.
A strong VPN will create an end-to-end encrypted connection that will connect you through the VPN to the VPN server. A VPN often includes a proxy that runs on your computer that intercepts all attempts to connect out (not just http over ports 80 and 443, but everything) and enforces connection rules (i.e. forces you to connect to a known server and disallows all other connections). So you will be sure that you are not being intercepted between you and the VPN server and being hijacked, hit with man in the middle attacks, etc.
What happens from there depends on your network. You could be restricted to only accessing other secure resources on your VPN, like company file servers or the accounting server. Or you might be able to go out on the broad web and download viruses, succumb to social engineering attacks and all other manners of bad things.
