What are the differences between VPN and dedicated server?

Perhaps, you mean VPS - virtual private server, not virtual private network.

Yup. VPS and dedicated server.

Hi,
VPS - Virtual Private Server

VPS is a solution, which is better than Reseller Hosting as you get root SSH access to the account. You need not worry of managing the server as the provider will help you to do it. You will have an allocated amount of server resources, unlike Reseller Hosting, which you have to share the resources with other users.

Dedicated server allows you to have root access to the server. Thus, you are free to do anything you like. You will use 100% of the resources yourself, without sharing with anyone else. Get a managed server if you are not sure of managing the server. If you are totally new to managing a server, get a server admin.

Hope that helps.

The main difference is that if I screw something up on my VPS, this will affect only my sites, not the sites on the other VPS' on the same physical server.

There are different virtualizing solutions which offer different degree of virtual independence. And it will never turn into an absolute one, as with the real dedicated server.

VPS is an excellent solution up to the $70-80 range. Beyond that, one should think of renting a physical dedicated server.