Redhat and Linux Mint have got quite a few differences, but with respect to the 2 main reasons to keep Linux profiles on Linux filesystems: no, none.
Keeping two (or more) Firefox profiles synchronized, even across different operating systems, is pretty easy: Use the Firefox internal "Sync" function. It allows you to synchronize: Add-ons, bookmarks, passwords, settings, history, tabs.
Beware: add-ons might pose a problem in case you use add-ons which are different between Windows and Linux. And yes, there are a few. Just check the default add-ons that Firefox brings along on Windows and those that it brings along on Linux.
Actually, in order to avoid conflicts with operating system dependent settings, file formats, I will never include add-ons and settings in the synchronization process.
I am not aware of a Thunderbird "Sync" function comparable to the function built-in to Firefox. Yet, it would be feasible, though involving some hassle to set it up right, to put the pure data files on a common filesystem. - Too much hassle for me. I simply created the same profiles twice. As a downside of my approach I will fetch all mails twice, once on the Windows side, and once on the Linux Mint side.
You might wait for other users' input. Maybe there is a better way, which I simply do not know.