Is there any benefit to running CiviCRM on a non-profts website server rather than its own server?
I have worked with many clients that maintain their website (usually Drupal, but WordPress and Joomla as well) and CiviCRM running on the same server, usually in the same domain instance.  I seldom have seen issues.

Now, of course, with a mis-sized or misconfigured server, you will have problems.  And if the website gets a huge amount of traffic, you will have problems.  And if your CiviCRM database has many hundreds of thousands of contacts, you will have problems. But these type of installs are few and far between.

There are definite benefits running a Drupal website and CiviCRM together.  You are easily able to embed contribution pages and event registration pages into a Drupal site. With Views Integration to CiviCRM you can pull information out of CiviCRM for display on the Drupal site.  With Webform Integration with CiviCRM you can build robust webforms in Drupal updating CiviCRM.  

Ultimately, the key is properly sizing and configuring the LAMP stack, Drupal, and CiviCRM. (Additionally, try to install on a VM rather than a VPS. You want full access (ala root/ssh) to tune things up.)