In terms of the HMD itself, there won't be much between the two. Anecdotally it seems that the Rift feels slightly more 'premium' in its build quality, and is slightly more comfortable but there isn't much in it. Image quality is said to be similar.
The two solutions use different tracking methods. Rift uses a system called 'Constellation', which requires you to plug a camera into your PC and that tracks infra-red LEDs (which flash a code so that the software can determine position and orientation). The headset can be tracked 360 degrees from a single camera and is said to have sub-millimetre precision. Vive uses a system called 'Lighthouse', which doesn't require you to plug anything into your PC. It comes with two lighthouse stations which are small boxes that effectively contain a laser and a motor and that continuously emit a pattern of light. This pattern is picked up by sensors on the HMD which uses it to determine position and orientation. This again gives sub-millimetre precision.
The lighthouse boxes communicate wirelessly (but require line-of-sight) or can be connected together via a wire. They only need to be powered, which means that they'll be easier to position than the Constellation cameras which need to be connected to your PC (and so may require USB extenders). Constellation processing is done on the PC rather than HMD so will use some CPU time, but it's said to only be a couple of percent per camera. In terms of tracking accuracy they're said to be similar.
Vive comes with motion controls at release. Its controllers are large wands with buttons on the front and side, a trigger on the back and a trackpad. They look a bit odd but their design is intended to reduce the potential for occlusion. Rift's motion controls (Touch) will come in the second half of the year and are a completely different design. They wrap around your hands, and have a capacitive element so the software knows when you've lifted your fingers, making it possible to point or do thumbs-up in VR. Touch will come with a second Constellation camera.
The big difference between the two systems is the type of experience that they're targeting. Rift is initially targeted at seated or standing experiences and comes with an Xbox One controller so that developers have a standard control method to target for their games. Vive can do seated and standing, but their differentiator is 'room-scale' where you're encouraged to get up and walk around in VR. Oculus don't officially support this configuration, but have said that there's nothing stopping you from doing it.
Vive has a camera built-in for their chaperone system. Chaperone has two components: the first is a simple boundary and when you get near that boundary it'll show up as an overlay on your display. This is to stop you from running into walls. SteamVR will support this level of chaperone on Rift as well as Vive. The second allows you to turn on an overlay of your surroundings, so you can see the outline of people and objects in your environment. This will allow you to see where your chair, desk, coffee, cat, etc. are. All the processing is done on the HMD so it's unlikely that there'll be a full passthrough mode but there are opportunities to use it for things like SLAM which would map your environment and project it into your game.
So... the real downsides of Rift are the lack of offical room-scale support, the number of USB ports required and unknown cost of the motion component. The real downsides of Vive are that there isn't currently a lot of content for its differentiating components, and that's something that Oculus have been very vocal about -- it doesn't matter how great a device is, if there's no content for it, it won't be a success. So with Oculus expect that when Touch *does* get released they'll do so alongside a bunch of quality content. HTC and Valve don't seem to be doing the same.
In conclusion, if you have the space and if room-scale is something that you feel you need from your VR setup, Vive would seem to be the one to get. If you're more interested in the seated / standing experience, there's likely to be more immediate content on the Rift, and Rift is (in the short term) cheaper.