I switched my current project from Three to Babylon about two years ago. At first it was just to experiment, and because I found Babylon's API way more intuitive.

But the main reason I've stuck with it is that Babylon has a compact, active set of devs (I've made bug reports and seen the fix in the next nightly build multiple times). So it's been really easy to get help and get bugs fixed. Three has way more people working on/with it, but the flip side of that is that it's way harder to find help, unless you know a good forum where some experts hang out, or similar.

The other big difference is that Babylon is more of a "game engine" than a "3D engine" - it has built-in subsystems for stuff like bone animations, audio triggers, etc. I imagine Three has similar stuff, but I believe they'd be plugins or modules that you'd need to manage separately. I don't actually use any of these Babylon subsystems though, so I can't say if they're useful or not.

three.js seems to be more well known/established than babylon and I had initially picked that for playing around. However, I got to go to a session on babylon at a microsoft conference and it was given by one of the authors and was pretty impressed. I had a freelance contract that needed some webgl and went with babylon and i'm glad i did. It better performance and the object names and concepts seemed easier to follow and a little less obtuse. IIRC, the presenter said that they are extremely diligent about working through bugs and issues on github and that three.js had quite the backlog. It helps that babylon is essentially being underwritten/subsidized by MS, where as three.js is more your typical open source project.

If you've invested tons of time working in three.js, then it may not be worth switching over. But if you're dabbling, it is definitely worth checking out.

However, all of this was about 2 years ago, and you never know what has changed.

FWIW (probably not much) I've looked at both briefly, to build a tool with a 3d component. I know nothing about graphics (basically flunked it at uni) but Babylon seemed much more approachable than three for basic stuff (a box and a camera). However there is a lot more written about three than Babylon.