I suppose that there is a small performance gain to be had by using separate files that are kept to as small an amount of code as possible (less code to compile on each page request). However, that gain is probably only measured in a few milliseconds at the most, so unless optimum performance is a high priority on your site, I would go with whatever method makes it easiest for you to maintain your code.
As I mostly do things now in an OOP way, I can visualize what you described as a single class file with separate methods for each functionality. Then the main script would instantiate that class and call the applicable method depending on the value from the URL query string.