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For details the discussion linked by Henry Cohn and the things linked there are great.
If you prefer a quick answer: At the time of creatiion of MathOverlow there was no stackexchange network. So, at that time there was no choice but to be independent. Whether now a change should be made, is ongoing discussion.
The history is actually described fairly well on Wikipedia.
Just one minor point:
I see no harm in having more than one mathematics stackexchange website, one for more research level based mathematics and one for more general mathematics questions
You may not see the harm, but there are plenty in the SE network who do not like having the split-level sites. MathOverflow would have to fight very hard to exist if it were started from scratch today.
@Andrew
Well, to be fair, they did agree to at least consider a research level physics Q&A even though there is already a general level physics Q&A.
Kaveh, I chose my words carefully. I do not ascribe any particular position to the SE company (though I have read disparate comments from various people in their employ) but to a group of active participants in the network who have made their position against such splits very clear. That there is a proposal for a research level physics Q&A is not evidence that this group does not exist! Indeed, if you look at the history (which I'm sure you know very well) then you will see plenty of activity by people trying to close it down. For those not familiar with this history, start at http://discuss.area51.stackexchange.com/q/2220/800 and read some of the linked discussions.
My own view is a bit complicated. I'm not actually sure that split-level sites can work, but actually I think that it is the lower level site that is contrary to the SE ideal. But the problem is that there are a lot more "non-experts" who will support a lower-level site than there are experts who will fight for the higher-level one, so the "democratic" method in use at the moment is biased to the lower-level sites, and once one of these gets created then the "no duplicates" brigade uses the existence of the lower-level one as an argument as to why the higher-level one shouldn't exist. (If anyone's actually interested in what I think, I've written it elsewhere.)
@Andrew,
Fair enough. Area51 is a quite strange place, everyone has an opinion and lots of them think that they should vote even when they don't know anything about the topic and are not using or going to use the site.
On the other hand, see this also which is on meta.physics.SE.
Btw, thanks, but I have already read your post. :)
Kaveh, I am not an avid follower of SE in general. But, I do not quite understand your 'on the other hand' link (but perhaps I missed something). What I mean is: what I mainly noticed there is that Robert Carvaino, a general community manager (if this is the right word), argues against the 'split'; among others on the grounds that there will be a competition for new users. (Which speaking from math and extrapolating from me a bit is not really and issue and actually the opposite seems true to me.) So, it seems to me this is an example of what Andrew means.
Fair enough. Area51 is a quite strange place, everyone has an opinion and lots of them think that they should vote even when they don't know anything about the topic and are not using or going to use the site.
Exactly the problem! I don't understand why SE insist on the process being so democratic.
On the other hand, see this also which is on meta.physics.SE.
What quid said.
Btw, thanks, but I have already read your post. :)
I'm well aware that others will be reading this and I posted that for others, not you. I know you're aware of that article (though I didn't know until now that you'd read it). I put things there when I think I might end up repeating myself, or when I feel that writing them down will help me clarify my own thoughts, and then link there rather than repost here (particularly when I don't know that others will actually care what I think).
@quid, @Andrew,
Let me explain my "on the other hand": we were talking about how other users feel/think about the two level split, I think votes on that meta.physics.SE post show that users (even when they are not researchers) act more reasonably when they are familiar with the topic and realities and are not defending an ideological position. (ps: I think the concerns raised by Robert were not unreasonable, but they were addressed convincingly).
I think there is a place for MSE as there is a place for MO. As someone wrote, MSE is teaching like activity, MO is for research. There are researchers and mathematicians who like to spend some part of their time teaching undergrads (but combining research and teaching undergrads might not be a good idea). My feeling is that having a research level site makes more researcher interested in also answering questions on the general level site from time to time.
Kaveh: thanks for the explanation. Regarding your last sentence, I completely agree; this is basically what I meant with 'the opposite'.
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