I've found that the JCE editor behaves differently in Firefox, as opposed to IE. Some of the differences are on the subtle side, others more obvious:
Paste:
* Firefox -- Opens a new window, requires ctrl-V
* IE -- Accesses clipboard directly, no ctrl-V required
Hyperlink editing:
* Firefox -- Sometimes, if no text is selected it acts like you are inserting a new hyperlink (all fields empty), even if the cursor is positioned within an existing hyperlink; sometimes automatically selects all text within the hyperlink
* IE -- If no text is selected, but the cursor is positioned within an existing hyperlink, consistently retrieves the hyperlink information from the surrounding area (text, image)
Edit/insert a hyperlink on an image:
* Firefox -- Have to use buttons, no editor-specific context menu
* IE -- Can use context menu
Edit/replace image:
* Firefox -- Have to use buttons, no editor-specific context menu
* IE -- Can use context menu
Toggle guidelines / view invisible elements:
* Firefox -- Guidelines / invisible elements always visible, toggle doesn’t do anything
* IE -- Guidelines / invisible elements never visible, toggle doesn’t do anything
HTML editor:
* Firefox -- Small font
* IE -- Larger font
I've also caught some other odd behavior, but it seems to be intermittent (or I don't know how to trigger it): In particular, I've seen Firefox create <span></span> tags to assign classes, with the same class assignment in an <a></a> contained in it. That's probably fine, because some browsers might ignore one or the other; but I've also seen redundant, nested <span></span> tags. None of this will cause a problem on your pages, but if you go into the HTML editor they can make things harder to edit.
I don't know if IE does the same thing, since I only rarely use it.
I haven't had (nor am I likely to have) time to test other browsers.