It’s important to understand that there are no “direct” SEO benefits to hosting on a dedicated server. In other words, search engine spiders don’t check if your website is hosted on a dedicated server in order to give boosts or penalties based on what they find out.
There are quite a few indirect SEO benefits associated with dedicated servers though. First of all, you don’t have to worry about your website being in a “bad neighborhood”. The main problem with shared hosting is that a lot of websites, sometimes even thousands, end up being hosted on the same server/IP.
Unfortunately for the people who only employ legitimate approaches to search engine optimization, not everyone is willing to play fair. There are webmasters who are always on the lookout for the fast way to success and it’s safe to assume that out of hundreds or thousands of websites hosted on a certain server, the chances are that at least a handful may be “problematic” from that perspective. In other words, even if you do everything right, there’s a chance that you’ll have to pay for other people’s mistakes.
There’s also the issue of page loading and page refresh speed which needs to be taken into consideration. Algorithms are constantly changing, and some search engines take page load times into consideration; at least to a certain degree. It makes perfect sense if you stop for a moment and think about what search engines are meant to do. Search engines all want to display the best possible websites when someone searches for a certain term and fast page load times are important when it comes to the experience of your visitors. Of course, there are other factors which also are taken into consideration and most of them have more weight than page load times. But still, having an edge, however insignificant it may seem, is always a good thing, especially if you’re targeting terms which are extremely competitive.
But, the most important thing that dedicated servers put on the table is complete control. It’s your server and you can use it as you please. This means that you have more control over the software that you’re using and that you can tweak absolutely everything.