OK, where to start...the PS3 vs. Xbox 360 battle was defined by multiplayer gaming. XBL and cross game chat made the Xbox THE place to play multiplayer games. This gen, Sony has improved their server infrastructure and has added cross-game chat, so on that, the playing field has leveled.
Next, there was VERY little difference between the multiplatform games...they ran at the same resolution, and the difference in frame rate was negligable. Sometimes there were bugs in one version and not the other...but for the most part, games played IDENTICALLY on the 360 vs. the PS3...aside from the factors I mentioned in the beginning.
This generation, we are seeing a fairly dramatic difference in the games. Most Xbox games run at 900p or less, and most PS4 games run at 1080p. Some games, like Tomb Raider or Sniper Elite 3 have 1080p resolution on both platforms, but then the frame rate suffers on the XB1. These are NOT subtle differences! They are obvious to anyone playing! If your framerate drops below 30fps, it's noticable! If your screen tears because they have vsync disabled, it's noticable! In comparison after comparison, news article after news article, the XB1 comes up short. Fans of MS try to minimize these shortcomings, but they are there, and people ARE paying attention...as reflected by sales.
The other mistake people make is to say "well the PS3 came up from behind, so the XB1 will also!". This couldn't be more wrong. The PS3 was able to come up from behind due to the strength of the Sony brand (especially internationally), the number of exclusives on the PS3 that were NEVER available on the 360, and the fact that many bought it strictly for use as a blu-ray player. Last gen, it was common for gamers to buy both platforms...the PS3 for the exclusive titles, and the Xbox 360 for multiplayer gaming. THIS generation, the LESS POWERFUL console with FEWER GAMES is the "underdog". There is not a good reason this gen to own both consoles, either, as the multiplayer experience is just as good on the PS4 as the XB1 and the number (and quality) of the Xbox One exclusives are just not there to justify owning one.