With some recent updates, the Xbox One is the better game console for cord cutters by a wider margin than it ever has been. Both consoles have several streaming apps and can play Blu-rays or DVDs. The Xbox One has always had a slight leg up on the PS4 when it comes to the number of streaming apps available. Some new services have launched in the last year and the Xbox One has received some new features through a software update that make it even better for cord cutters and streamers. The Nintendo Wii U also has some streaming features, but has not sold nearly as well as Xbox and Playstation, so it's not included here.
The Xbox 360 required a Live Gold subscription to use streaming apps, but this is no longer true with the Xbox One. Both Xbox One and PS4 have apps for the major streaming services such as Netflix, Hulu Plus, HBO Go, and Amazon Instant Video. Xbox One has a few other popular apps PS4 does not: NFL, ESPN, and Sling TV.
The NFL and ESPN apps have video clips, fantasy football stats, and more, and can be 'snapped' to the side for a split screen view while watching a movie or playing a game. The ESPN app also has a ton of live sports on watchESPN. Anything on ESPN3 is free with select internet providers, but everything else requires a pay TV subscription. Sling TV is a new live streaming TV service that includes AMC, ESPN, and several other channels for $20/month. HBO is also available through Sling for an additional $15/month. Try a free Sling TV trial.
Playstation just launched Vue, another new live streaming TV service. However, I don't recommend it because it costs $50-$70 per month, is only available on PS3 and PS4 right now, and is missing ESPN and a few other big channels.
Watch and pause free live TV
The Xbox One was designed to be used with a cable box to watch live TV, but it wasn't well executed and doesn't seem to have taken off. However, with a recent software update the Xbox One can be used to watch live TV in HD with a separate antenna and tuner. The update includes some DVR features, with the ability to pause live TV for up to 30 minutes. This isn't a proper DVR, but a nice feature and perhaps additional DVR features will be added in time. Get more details here.
Naturally, games themselves are the most important thing to consider when buying a game console. All things being equal, the Xbox One is the better choice for cord cutters and people that stream a lot of video. It has more streaming apps, the ability to watch and pause live TV with an antenna, and has some neat system features such as Kinect voice control and split screen viewing. The Xbox One starts at $349, $50 less than the PS4.