The complexity of installing/running software usually far outweighs any small gains by choosing one platform over another.
What I mean -- if you are comfortable with Windows or Linux - stay with it. Don't switch from one to another just become someone tells you the one you're not using is faster/better.
In the long run, knowing how to manage one system extremely well is better than only knowing a little bit about all of them.
Specifically, if you know windows really well, keep running MySql on Windows. If you know Linux well, stay with that.
If you have performance problems then learn how to tweak all the settings on the platform and operating system you are using.
Then, if you need more speed - as much as this sounds simplistic, just throw more hardware at it.
It is a lot easier (and in the long run cheaper) to just throw more ram in the server, use a faster hard disk/controller, and use a faster CPU than to switch architectures.
Although not as elegant, because servers are cheap and every few months are getting faster and cheaper, the overall cost to throw more hardware at a decent performing system to make it faster is the simplest approach.
You do need to have your system reasonably under control. A badly configured system will not improve if you throw more at it - but otherwise this is actually a pragmatic approach that works for the vast majority of sites/servers that don't need to squeeze every last ounce of performance out of the existing setup.