If your site is only standard html/php/mysql or what have you, I wouldn't worry about separate servers...but if you are using something which runs code, processes, sever apps, ect, it may be worth it.
Alot of big sites now use things like plone which aren't just a set of files read by a default server app like apache, they are their own program which dynamically configures content. Changing your site may mean changing code int the environment of your server or operating system, so operating alternate sites for your two different ones is a good idea.