I've heard from various sources that some things are inconsistent between the desktop and metro versions of IE in terms of remembering open tabs, preferences etc.
Where a feature exists in both, what is synchronised between the two and what isn't?
I can't speak to the feature differences. (As far as I know, there aren't any feature differences except for the obvious UI changes, and the sandboxing/security things that the majority of users won't have to worry about.) But, regarding the "synchronization" between the two:
Generally I suggest choosing one, and using it consistently. If you know the Windows 8 gestures and/or have a touch screen, I suggest the RT version. But if you are in Desktop mode most of the time anyway, maybe the old, desktop UI would be a better choice.
(Remember, you can dock the "RT" version so it doesn't stay full screen. So that shouldn't be a significant reason to choose one over the other.)