Why can't non-research level questions simply be posted on mathoverflow with a special tag? Then people who wanted to answer MSE-type questions would just limit themselves to that one tag. I am asking purely hypothetically, of course, as I know the sites work well the way they are.
While the topics are different, a blog post that Robert wrote some time ago explores the reasoning behind 'splinter' sites, and under what reasons we'd support them. There are also technical considerations to speak of, MathOverflow was an autonomous SE 1.0 site not connected to the Stack Exchange network in any way for a very long time. They've been part of the network for less than two years now.
We'll get back to that later, let's talk generally.
It's not just math, why do we need:
There used to be a site to talk about theoretical physics when we already had a site for physics.
I won't get into those computer science people, they have special mind powers and I fear retribution.
.. it comes down to showing enough interest, support and expertise in a topic along with reasons that it clearly wouldn't work as well as it could as a tag in another overlapping subject.
People that use MathOverflow are purely interested in research-level stuff, and while tag filtering does work well enough to keep out 'noise' - there are some other things to consider:
Communities need to agree on how they're going to handle certain things, elect their moderators, and use their votes to keep the site healthy. When you have two distinctly different groups of folks sharing the same roof - this gets difficult.
We need to be able to promote and encourage sites as a single community. This is very difficult when you have two groups with distinctly different needs, goals and preferences.
We prefer not to fragment groups of professionals or academics when it's clear that they have much more in common than not, and we're very careful to not do so unless it's a clear necessity.
Now, going back to MathOverflow - there's no way they would have joined the network as part of a more rudimentary site, it's just not in their topic scope at all and the experience would have been awful for everyone involved. So, let's pretend that MathOverflow was born out of Math SE users saying "We do research level stuff, and I need a place where everyone else participating is doing the same thing in order for that place to be useful to us." - we would have agreed.
I love our coffee site. Even though I'm an avid home cook, I like pulling a double shot of espresso, sitting down and seeing a front page entirely about coffee, it makes me want to talk about coffee even more.
And that's why we do it.