Yes, I know, that's a cop-out, but not really… There are benefits and drawbacks to both, and what your are designing and who you are designing for is just as important as the design "style" ("Flat" or "skeuomorphic") you choose to use.
"Flat" design is a good approach if you want to focus on and provide clean simple UIs, but "Flat" design can sometimes be harder to understand because the familiar visual cues that users expect or look for aren't applied. This is especially true when you are redesigning something that has a "history" with the user. Exhibit A: The tiled UI in Windows 8 strips away the standard conventions users have "learned" over the years, increasing the potential learning curve and cognitive load for the user.
Skeomporphic design provides real world visual metaphors to the UI that can help the user better understand how the application works, thus reducing the time needed for users to "pick up" how the app works (a good example of this is the Notes app in iOS). The flip side to that being useful is when it is done as simple visual "skinning", which does nothing to add value to the user and can come off as garish and out-of-touch. Exhibit B: The "reel to reel" player view in Apple's Podcast app. How many users alive today ever even SAW a reel-to-reel player in the real world before? Heck, the only reason I recognized it is because I worked in radio in college. It's a visual treatment that is, at best, "neat" – at worst it's a hipsteresque affection.
You can do good work using both styles, and design trends moving away from Skeuomorphic design doesn't mean it's "bad." Bad design can be done in ANY style or approach… Skeuomorphism is just not the trendy style it once was, but it's no better or worse than "Flat" design.