Overall, the PS4's interface feels zippier than the Xbox One's. Games install quicker and moving around menus is a much more seamless experience. It's by far an easier system to navigate, as opposed to the Xbox One's sometimes confusing presentation.
Sony's answer to backward compatibility is PlayStation Now, a service still currently in beta that allows PS4 owners to stream a game over the Internet. That said, your experience will vary depending on your Internet connection. Suffice it to say, playing shooters and other "twitch" games on PS Now isn't great. We're also not in love with its current pricing structure.
That said, while it's far from perfect, the Xbox One doesn't have an answer to PS Now.
Aside from a zippier all-around experience in the system software, the PS4 tends to install games quicker than the Xbox One. There's also some evidence that multiplatform games play better and run in higher resolutions than they do on the Xbox One. In some cases, the PS4 will also play at a higher frame rate than the Xbox One.
Game broadcasting and social sharing
The DualShock 4 controller has a button dedicated to broadcasting and sharing options. The whole feature set is wonderfully tied into the fabric of the system and makes doing so fairly painless. Players can instantly snap screenshots, tweet photos and broadcast gameplay to Twitch (a free online streaming gaming video service), all within a few clicks.
PS4 owners can also save these videos and screens and put them on a USB drive, edit them on the PS4 or upload them to YouTube (thanks to the aforementioned 2.00 firmware update).
Unfortunately, publishers can block the ability to share content -- something Activision is doing with Call of Duty: Advanced Warfare.
Sony has committed to bringing popular independent games to PS4. While a lot of these titles have previously been available for PC, games like Nidhogg, The Vanishing of Ethan Carter and Broforce (among many others) will only see console debuts on PS4.
User-accessible hard drive
The PS4 ships with a 500GB 5,400 RPM hard drive, but users can easily swap it out for a 2.5-inch SATA drive with a larger capacity or a SSHD or SSD for potentially increased performance. The Xbox One, by comparison, doesn't allow user-upgradable hard drives.
DualShock 4 controller
The DualShock 4 is the best PlayStation controller yet and features a front-facing touchpad that can also be clicked. Players can bring their own headphones and plug them directly into the controller so they don't disturb the neighbors during nighttime gaming.
The controller is very comfortable and can be charged with a Micro-USB cable. The only real downside is the battery: unlike the Xbox One controller, the PS4's can't be replaced.
Here are the areas where the PS4 could use a little work:
Media apps: Good, but slightly lagging behind the Xbox One
The PS4 offers mainstay media and entertainment apps like Amazon Instant Video, Netflix and Hulu Plus, but is noticeably missing apps that the Xbox One does have, such as ESPN, Comedy Central, Fox, Fios and others. And weirdly, neither the Xbox One or the PS4 have HBO Go, even though the older Xbox 360 and PS3 do.
There is support for sports, though -- PS4 owners can use MLB, NBA (only on PS4), NFL Sunday Ticket and NHL apps.
Sony has also teased an online TV service, which we've yet to see, though it was promised to hit before the end of the year. Details are vague, but it looks to be a cable-like video streaming app -- think Netflix or Hulu, but with live TV channels.
Media playback: Not yet as good as the PS3's
Aside from the built-in video and music apps, PS4 owners can't play their own video media on the console, unlike the media player capabilities now featured on the Xbox One. They can, however, play music from an attached USB drive and listen to it while playing games.
Unlike the PS3, the PS4 has no support for DLNA -- which means you can't stream photos, music or video from a PC on your home network.
PlayStation Plus: Worthwhile, but not as good as a deal as on the PS3
While PlayStation Plus gives PS4 owners a decent game collection out of the box, it's much more expansive on the PS3. This will likely improve as the PS4's library continues to grow.
PS4 vs. Xbox One
There's not likely to be a definitive winner in the new generation console wars. While the PS4 had a clear advantage at launch, that edge is slowly evaporating as Microsoft has worked feverishly to undo most of the Xbox One's original missteps. Both consoles are now similarly priced and offer a lot of the same features.
While the Xbox One has had to play catch-up for the last year, it's entirely possible the pressure will eventually get to the Sony camp as well. The PlayStation 4 has yet to see a price drop, whereas the Xbox One has tweaked its pricing structure a few times.
Right now the PS4 and the Xbox One are neck and neck with exclusives -- though the PS4 also has a better range of digital-only titles. But taste in games is always subjective; either those games will appeal to you, or they won't. Each console manufacturer has made exclusivity deals with various developers, so the sad reality is you're going to miss out on something great no matter which platform you choose.
You might read about the PS4's specs trumping those of the Xbox One, but it's important to keep in mind how that translates to actual results. You'll remember that the PS3 was originally poised to be a massive powerhouse that would leap past the Xbox 360, but in reality it didn't perform much better. You could even make the argument that most multiplatform games played smoother and looked better on the Xbox 360. That said, at the time of this writing (and having considered most of the multiplatform games currently available), the PS4 does seem to perform slightly better than the Xbox One.