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: 
You can't please everyone
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Most changes or additions will leave some people behind. They may catch up later, they may hate you forever. Shoot for net gain in the experience. If you avoid negative feedback, you avoid progress.

It helps to keep a destination in your sights. Focus on an established list of goals for the long term vision of the product and the tactical changes of a sprint or release. Using something like [the HEART framework](http://bit.ly/16TKfSD) is a good way to keep your UX on track.

Don't ignore the complainers
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Take your inevitable complaints seriously and evaluate them for opportunities. Mapping complaints back to your goals will help you decide if you've missed the mark or just need to help bring a subset of users along to the new way.

When you see opportunities in feedback, do a little research. I've found that most complainers love to be heard: follow up with them and do an interview. If you find potential, develop a job story around it and add it to the backlog. When that item fits the goals of a design sprint, you'll roll it in and show that user that you're listening -- as long as you don't take too long ;)

Account for roles and personas when possible
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You may notice clear distinctions in the responses that point to persona or role groups. If you can provide an altered view for this group, you may open up a whole new territory for your product. With the flexibility of software and power of hardware today, faceted UX isn't as prohibitive as it used to be.
Accommodate both! In this case, your users told you exactly what they were missing in the old version (in this example, easy copy/paste). So create a new way that meets both sets of requirements.

Generally speaking, say the old way offered Features A and B, and the new way still supported Feature B, lost feature A, but added feature C. Users said they like C but missed A. So design a widget that offers A, B, and C!

In this example, when the user pastes a comma-separated value, you could automatically transfer the part after the comma to the second box. Also, if they are typing in the first box and type a comma, automatically transfer focus to the second box as they continue typing. You can see this behavior in a lot of phone number entries on forms. Perhaps this would satisfy all your users.