These are genuinely different products and picking the wrong one causes avoidable work.
A game server
A managed instance of one game with a control panel built around it: console, file manager, backups, scheduling and startup settings. You manage the game, not the operating system. This is the right choice for the large majority of people running a community server.
A VPS
A blank virtual machine with root access. Far more flexible — you can run anything, including your own panel node — but the operating system, security, updates and backups are yours to handle.
- Want a Minecraft server for your community? Game server.
- Want to run a Discord bot, a website and a database? VPS.
- Want to host several game servers yourself with your own panel? VPS or dedicated.
- Do not want to administer Linux? Game server.