While I never owned a 360, I just felt like Microsoft just didn't care about the core gamer by the time I got my PS3 in 2011.
Microsoft had a focus on media (well before the whole X1 Media Box-thing that was said at the X1 unviel), larger focus on Kinect software and the "safe" franchises (Halo, Gears, Foza, Fable) for its first party line up, (a personal thing) them butchering the talented developers at Rare (so many great concepts canned
...), and having online multiplayer cost money when it was free on the other consoles at the time (yes, you have to pay for PS+ for Mulitplayer on PS4, but you get so much out of the service with the Instant Game Collection and Steam-level sales).
I just feel that for people looking to get next gen system at this point, the PS4 looks to be the great long term investment; cheaper price point, garented library of many solid 1st party exclusives, the better running/looking multiplatform games, great services like PS+ and the coming PS Now, cross-buy with many PS3/Vita indie games and more. With the X1, it is a good short term investment, as even though it costs more money, you do get some more first party games and "exclusives" (Titanfall and Garden Warfare are also on PC) while you have I believe a more stable online service (even if you pay for online, at least the online has solid support; why else did many gamers pay for Live on 360 if it didn't offer great connections?).
Either way, both look to be great systems with different features and games, but from the long term standpoint, I personally think that the PS4 is going to be the better of the two systems, thus explaining why so many gamers are "jumping ship" so to speak to from Microsoft to Sony.