MathOverflow is probably more popular than Math.SE counterpart. Are there any other examples of third-party sites powered by SO engine that have become more popular than SE counterparts?
http://biostars.org/ covers just a small subset of biology.SE, and touches on computational science too. It's a third-party site.
Until mid 2012, it ran on the SO engine, and is reasonably big given its very specific subject area: it had 4635 questions and about 2400 users while running on the SO engine. At that time, the biology.SE and computational science.SE sites were both very new and in beta, so one can't make a good direct comparison of their relative popularity.
Other than MathOverflow, Biostars was one of the last sites outside the SE network that ran on the StackExchange engine.
MathOverflow.net does belong to the SE network now.
BioStars is for bioinformatics and computational biology. It formerly ran on the SE 1.0 engine. At some point, they had the choice of licensing the SE platform for $25,000 I think. I don't know if that was annual or one time. Alternatively, they were offered the option of joining SE through the Area 51 "proving ground". That's my description, and is not meant pejoratively! They could not reach consensus among themselves, though. A third alternative was worked out, where the site remained online but static, as a repository of still useful information. It seems that BioStars.org has been resurrected and is quite active again! It is sponsored by MIT and UPenn etc. and is open source.
OSQA is a free, entry-level Q&A system from the makers of AnswerHub, the market-leading professional Q&A platform. OSQA is a great solution for smaller sites with limited needs.
OR Exchange functionality resembles SE, however, note this, on the OR Exchange FAQ:
Who are you really? This was started by Michael Trick, but is now owned by the community, and currently sponsored by INFORMS.
Michael Trick, PhD is well-known OR professional, perhaps professor as well. INFORMS was my professional organization when I was more active in that field. It is analogous to SIAM, ACM or IEEE.
So OR-X is a special case of OSQA.
More about OSQA
OSQA certainly has the "look" of SE at first glance! See the meta site
My next question was, "What is this Answerhub that is associated with OSQA?" OSQA is free Q&A software licensed under the GPL, whereas Answerhub is the professional product, for enterprise presumably.
I was curious about this as well. Here are a few I could find:
- BioStar - bioinformatics/genomics (running a SE clone)
- MathOverflow - research level mathematics (uses Stack Exchange 1.0)
- MetaOptimize Q+A - data science
- OR-Exchange - operations research 
- aiqus - online learning community  (running OSQA, a SE clone)
 I have no idea what that is.
 Recently a bunch of free online classes have been popping up around the web, such as Coursera, edX, and Udacity, as well as a couple independently offered coursers. This community seems to encompass all of those, among others.