Question: Getting clear user feedback is great. But what do you do with the feedback when different people, or groups of people, want different things? 

I have encountered this situation a few times recently.

One example, while testing a new product release: Some users liked the new separate input boxes for x and y coordinates, as the previous version of the software had a single box, with a comma separator. Now they didn't have to type the comma in.

But other users disliked the move to the dual boxes, because it increased the amount of work involved in copying/pasting co-ordinates between boxes. 

I'm not asking for a design solution to this single box vs. dual boxes problem :) I'm asking, more generally, what methodology do you employ to arrive at solutions, where there are two or more opposing feature requests with seemingly equal merit?

Answer: 
Try to distinguish between *what* users want and *how* they want it done.

Taking your example above, users wanting one vs. two input boxes is all about the *how*. The *what* is being able to paste comma-separated coordinate pairs vs. not having to press comma. (Or, for some users, being able to simply press comma rather than having to click a second input box.)

In the above example you could resolve that in the following way:

* Implement two separate boxes.
* When a comma is entered in the first box, transfer focus to the second box
* Implement copy and paste buttons that will copy the two values to the Clipboard (as comma-separated values) and paste a comma-separated pair from the clipboard, splitting it correctly between both boxes.

Implementations will, of course, vary heavily depending on the situation you're facing. However, many apparent conflicts between requirements will go away when you look at the what/why rather than the how.
I am programmer learning a bit of UX so I may get beat up by UX purest but I capture metrics on user productivity. If a top performing user asks for a tweak I will take that over a low performing user. I am in an environment where contract data entry use more than one product. Too often a low performing user will state the (my) product is the problem. I will show the customer paying the bill that the top 1/4 users are doing over 1/2 work (and they always are) and those are not the people complaining. I am not saying don't accommodate the low end users. But don't sacrifice the high end users. A user that has performed a task twice is going to have in uninformed opinion. Yes make it more intuitive. But the top performer that says it saves me a key stroke is who wins in my book (as that is what the people footing the bill care about with my product).