Hi Kyra,
Typically a Managed Dedicated Server is hardware that a company will manage for you that you have all to yourself. You don't share the hardware with anyone. You usually have more control over what is installed.
A Managed VPS (Virtual Private Server) can be a vritual server that is sharing hardware with other virtual servers. So the performance might be less. You often have less control over what is installed, because installing some applications might not perform well and impact the other users poorly.
Hopefully others can chime in more... I wouldn't rule out a dedicated virtual server, which is slightly different than the other two listed above. Sometimes it depends on your budget. In my opinion, Managed Dedicated Servers usually cost more but run better than Managed VPS. ITs dependent on a lot of factors, though.