A Game Server is only the instance of the server you bought. This means it is limited to only running that particular game with the amount of slots you've paid for. The advantage for the hosting company is that those are easily manageable with no additional license fees.
A VPS (Virtual Private Server) is an own instance of a whole server system, including an own operating system, where you get full root/admin access.
Server virtualization isn't free. Also they're harder to manage, so extra administration hours are needed to maintain reliability and stability. And you can do whatever you want with it: Install more than one game server, voice server, a webserver,... so the extra stuff you can do with your server may be paid for as well. Usually VPS come with more disk space than regular game servers and server hard disks are pretty expensive.