There might be other implementations of the JSF base than the two that you've mentioned, but if so, I don't know them. I started out using Myfaces, but switched to Sun when it became more reliable. As far as things go, the 2 different implementations both cover the same standard, so they're supposed to be interchangeable. In any event, JEE servers have the JSF implementation integrated into the server (J2EE didn't). Once the implementation is provided by the server, it's pretty much a moot point except in cases where you have a need to override it.
The extension tagsets such as RichFaces, IceFaces, PrimeFaces, Apache Tomahawk, ADF, etc., etc., etc. are all extensions to the core and they neither conflict with, nor supplant/override the core JSF tags and classes. Although they may sometimes conflict with each other.
As to market share, I don't know. Locally, IceFaces is pretty popular, although my business is using RichFaces. We get a pretty even mix between RichFaces, IceFaces and PrimeFaces questions in this forum. Lately, RichFaces has been popular enough that people forget to mention what extension framework they're using, but that's hardly a scientific poll.
As in the case of IDEs, the really important question is "what are the people hiring in your town using?"
An IDE is no substitute for an Intelligent Developer.