A dedicated server is your own rented computer with a high speed internet connection. A VPS is a portion of a dedicated server that acts like a virtual dedi. In a VPS you have to share the network connection and CPU with all the other VPS' on the same dedi. Some providers guarantee a certain amount of CPU power per VPS. Others share it equally among the VPS. Most providers offer guaranteed memory to each VPS. They may also offer burst memory which is shared among the VPS'. However the dedicated server used for these VPS' is likely to be a powerful one with at least 2 CPU cores and lots of RAM. As an example the node of the VPS I am currently using has 8 cores. IMO you should stick to a VPS for as long as you can since they offer better value for money.
Stay away from VPS unless you personally know every single other person on the box. A VPS is just like a condo, if you know anything about realestate on the beach.