The large majority of web sites that you find online are hosted on shared servers. There are indeed benefits to hosting your web site on a dedicated server, as will be explained below, but for most people the significant jump in cost does not justify such a decision.
You may consider to get a dedicated server if (a) you host over 50 websites and/or some of your websites are really busy (e.g. because of busy forums), (b) you run scripts that require special settings for e.g. PHP, (c) you run your own billing gateway or other sensitive scripts that you need to make as secure as possible (by closing unnecessary ports and other security measures that would clash with the needs of other customers in a shared hosting environment, (d) you like to run your own nameservers, (e) you need to install server level scripts that are not being offered in a shared hosting environment, (f) you simply want to be in charge, and not endanger your site or email functioning because of possible abuses by other users on the same server.