I'm not exactly adding anything new here with my opinion, but after working on shared hosts for a long time, going to a dedicated server is a breath of fresh air. Admittedly, you have to have some UNIX skills (usually, unless you're running on a windows host which isn't all that rare I suppose), but the payoff is immense. Your site is almost always faster, and if you do anything involving dynamic code and databases (which is arguably 99.99999% of the net), you'll welcome it.
Just for PHP sites alone, being able to install PECL/PEAR modules, modify the php.ini directly, add additional vhosts and adjust the httpd.conf file for Apache and my.ini file for MySQL is insanely useful. It's especially hard using shared sites when you need to work with third party provided SDKs or APIs that involve open_basedir restrictions or other things.
If you have the capital and you anticipate your site ever needing some real power, I'd strongly suggest going dedicated as soon as it is feasible to do so.