Quora does not use MathJax; equations render as images on the site (and they are typed as
[math] ... [/math]). Stack Exchange uses MathJax, and is one of five MathJax partners, which means the level of contribution $\$20,000$/year. Quora is not among sponsors. I think it's important that MathJax development continues and the product (codebase and CDN) remains free to use. For this reason, I prefer to support a company that supports MathJax.
Stack Exchange makes its data freely available, through Data Explorer and for download. Quora says they might do it some day, and they've been saying that for a few years. I prefer to contribute to a company that does not hoard what was given freely. Mathematicians have a habit of putting their knowledge in a lockbox, handing the key over to a publisher, and then acting surprised when the publisher charges arm and leg for access. I think this was done enough times already.
Platform for development
At present, only Stack Exchange is a platform in the sense that one can develop applications that interact with it, using its well-documented API. (I made a couple of Chrome extensions that do that.) Quora does not have a public API that I know of.
Quora insists on full real names, and even shows user's occupation/position next to their name. I strongly dislike this practice, and not just because I prefer ever-changing and somewhat quirky display names for myself. I think that mathematical contributions should be evaluated on their own, with no regard to the author. I'd stick with system-generated names userNNNNN, but human memory isn't adapted to storing numbers like that. A randomly generated $k$-tuple of words would be better; I'd rather see correct horse battery staple than John Q. Public from Springfield, IL.
Quora's real name policy is the deal-breaker for me. And spare me from Facebook integration, for Zuckerberg's sake.
I tend to ask questions via Google search instead of posting them. By now it's nothing unusual to find a useful answer on MathOverflow or Math.SE among top results, even when the search is not restricted to these sites. I am yet to find one on Quora.