With Facebook's Video Calling, you can have a videochat with one person at a time. If you make a videocall to someone already on videochat, you'll get a notice that they're already in one, but you can leave a video message if you'd like.
With Hangout, you have a group videochat. In fact, it's been designed that way from the ground up. If you'd like, you can have one-on-one chats, of course, but the design approach is the more the merrier.
More features and controls
When you make a videocall in Facebook's Video Calling, you don't get much control over your call or extra features. In fact, essentially you get none.
That's not the case with Hangout. When you're in a Hangout you can mute your video or audio, do a text chat, and invite others. And tweakers will be pleased to know that there are settlings to fiddle with. In addition, there's a nice built-in troubleshooter as well.
The serendipity factor
You create a Hangout for a specific circle or groups of circles of friends on Google+, and can invite others as well. That means, though, that you'll never really know who will show up. The fact that you've created a Hangout gets sent out, but each time you do it, different people will show up. That makes the Hangout videochat more entertaining than Facebook's Video Calling.
Google is an engineering-heavy culture, and you can usually count on some oddball feature or other slipping into new services they introduce. That's certainly the case with Hangout. Consider this: In a Hangout, you can switch from a videochat with an individual or group to instead watching a YouTube video. Why would you want to do this? Frankly, I don't know. But if you're in a work-related videochat, it's nice to know you can cut out the droning and instead get to the important things in life, like watching a cat use a toilet.
For a more comprehensive comparison of the two services, check out my review, Hands-on: Facebook Video Calling vs. Google+ Hangout.