Both  provide a complete server environment on which you can run any  software, including games. From the perspective of the user, dedicated  servers and virtual private servers are similar, but which you choose  depends on the requirements of the games and the number of players you  intend to host.
Dedicated Server 
A dedicated  server is a physical computer in a data center. When you run software on  a dedicated server, it runs on the “bare metal”. The full resources of  the physical machine are available to the user and any software they  run.
Virtual Private Server 
A VPS is a virtual  machine. You can think of it as a simulated server running in software —  a hypervisor — on a dedicated server. That makes virtual servers  flexible: they can be resized or moved to a different physical server.
Server  hosting providers run many virtual private servers on a single  dedicated server, each of which gets a slice of the server’s resources:  RAM, storage, CPU, network bandwidth. When you lease a virtual server,  you choose how big that slice is.
VPS hosts often oversell  virtual servers, allocating more resources than the underlying physical  server has in the hope that clients won’t need all of the allotted  resources at the same time. Games need all the resources they can get,  so overselling can cause contention and performance issues that affect  gameplay.
Dedicated servers have resource limits too, but those  limits are clearly displayed when the server is leased. Server hosting  providers can’t oversell dedicated servers (although they might oversell  network connectivity).
How To Choose A Game Server 
Most  games that can be self-hosted provide guidelines about the server  resources they need. Check the requirements to make sure the server you  choose is capable of running the game with the number of players you  expect to host.
Add-ons, modules, and expansion packs can increase the amount of RAM and storage you need, so be sure to take that into account.
Cheap VPS
A  low-powered cheap virtual private server won’t cut it. Inexpensive  virtual servers — think $5-$10 — allocate minimal processing power and  RAM. They don’t provide the resources needed by even a simple game.
For  example, Minecraft requires a minimum of 2GB of RAM. The cheapest VPS  game servers provide 1GB or less. With 2 GB, a VPS can support 1 - 4  players with acceptable performance. More than five players requires 3  GB and more than 10 requires 8 GB.
High-Powered VPS
Some  VPS providers offer higher-powered virtual private servers with  sufficient resources to host games. They are often marketed as hybrid  servers: fewer virtual servers are hosted on the same dedicated servers  and each server has more resources.
Hybrid servers can provide  decent performance as a moderately busy game server. They are less  expensive than dedicated servers, but the prices of the most powerful  hybrid servers are in-line with similarly specified dedicated servers.
Dedicated Servers
Dedicated  servers offer the best game performance. They range from moderately  powerful machines ideal for small personal game servers with a handful  of users to the most powerful single-chassis machines available.
A  top-tier dedicated server may have multiple processors with dozens of  cores, hundreds of gigabytes of RAM, and several terabytes of storage.  They are capable of hosting hundreds of concurrent users without  breaking a sweat.
You can expect to pay under $100 per month for a  low-end dedicated server more powerful and reliable than most VPS’s on  the market. For a top-tier dedicated server, you will pay in excess of  $1000 per month.
A Word On Networking 
Network  performance is just as important as server performance. The most  powerful dedicated server provides a poor gaming experience if it is  hosted on a high-latency network.
Most hosting providers  advertise the bandwidth they provide with each server: the amount of  data the client can transfer in and out of the provider’s network. That  is important, but make sure that the network is also optimized for low  latencies (often referred to as ping rates by gamers).
In Conclusion 
Your choice of a game server should take several factors into account,
- The resource requirements of the games you intend to host
 
- The number of concurrent users
 
- Your budget
 
Generally  speaking, a dedicated server is the best solution but a high-end  virtual private server might be good enough, depending on the game and  the number of users.