Generally speaking, it goes like this: With a
NAT router, LAN clients' connections are forwarded directly to the WAN servers. With a
proxy server, LAN clients connect to the proxy server, and the proxy server starts the connections to the WAN servers on behalf of the LAN clients. You would probably use a proxy if you want more control over the
application layer, and a NAT router if you want to focus on the
network layer. Lots of people use both simultaneously, such as in cases where you want everything to be NATed except Web traffic. My explanation is indeed oversimplified, and I recommend you check-out the links I provided to get the whole story.