On games that are multiplatform the PS4 will be slightly better than the Xboxone, usually with the detail and framerates.. However it's so slight you likely won't notice. However as it is early, when games start to take full advantage you may notice things like Xbox needing more disks when PS4 is a single disk (as happened in the past), or not be as clean or smooth. It's early in the process still, it takes sometimes years for games to really push the limit.
So games like Call of Duty, Madden, basically all the huge cash cows that are just looking for money, you won't see much of a difference at all.
The difference, as noted above will come from the exclusives. Sony has made the PS4 about the gaming, because that's what it's roots are, and that's what gamers want. Gamers want to play games. Sony is more friendly to indie gamers, and all a result they will get exclusives that have more... heart I guess, put into them. Better stories, better graphics, better in general assuming you like the genre of course. Sony is friendly to them because even though they don't always make the most money they often make the best games. Naughtydog is a good example (Last of Us, Uncharted, etc)...
Xbox is trying to be THE box you put in your house, your DVR, everything.. They're about snapping to live TV with PiP (which makes no sense to me personally) and all that jazz. They focus is more on media in general, they want to be the jack of all trades, but not really master anything. Their online is better but a lot is the popularity that has pushed them to focus on that area.
Really what it comes down to though is what you want it for. If you want it to play call of duty with your friends, get what they get. If you want it to be the media hub and easily watch YouTube and stuff, get an Xbox. If you want the best games and the system best geared for that (with yes, the more powerful system) then get the PS4. They're marketed to different groups really. PS4 is for the gamer, Xbox for the guy who just wants it simple and easy.
Side note though, when my dad was working for EA Games and Activision, they hated having to throttle what could have been better games so the Xbox could run it. They often talked about putting it into the PS3 version anyway, but higher ups didn't like it. However a few games would give added bonuses to the PS3 only as an "exclusive" when it was usually in there to start with anyway and they had to scrap it to save space or make the game work on the Xbox.
Oh and the PS4 youtube app comes out in a few months.