Hello Robert,
I would always prefer to go with Dedicated box instead of a shared hosting package.
There are many factors which has to be decided while choosing for a shared/dedicated packages.
1) Application/softwares you are running on the website.
2) Daily traffic/hits and size of the website.
3) Web space and Bandwidth requirements!
4) Your budget/prices with particular specifications.
5) Last but not least your knowledge/skills with root access of the server, administrations knowledge, etc..
If your site is fairly small, then high-end shared hosting account is suitable for you. And if you need more reliability with optimal performance, premium bandwidth and need to offer game hosting than you can go with a dedicated box with good specifications
