There is a slight benefit when using a dedicated page file partition in that this pagefile (read: virtual memory) will neither fragment nor make other files fragment. However, if you defragment your drives regularly anyway, this should not be a problem.
Linux for example always tries to use a dedicated partition with a special filesystem as virtual memory (called swap). Many Windows servers are set up that way, too.
Windows circumvents this extra partition by mounting a file (pagefile.sys) as virtual memory file system, which might impose a very tiny performance decrease due to the aforementioned fragmentation problem. But outside the server world this should not be measurable.
Greetings from northern Germany,