They are two very distinct offerings linked by the generic-yet-fitting-in-both-cases term “managed” in their names. Managed Dedicated Servers is focused on your infrastructure. Managed Hosting is focused on your environment.
Managed Dedicated Servers ensure the uptime of your critical hosting infrastructure — including servers, firewalls, load balancers, etc. — and allow you to manage everything from the operating system up.
Managed Hosting is a complete soup-to-nuts service offering in which your dedicated team has an intimate knowledge of your applications and databases. This will maintain a continuous conversation with you to tweak your systems, scale your installation and plan for the future. If you have database issues or you find your application is loading slowly, the Managed Hosting team will work through these issues with you.