The way Mozilla worked and Firefox still works is there is a cross platform front-end implemented in XUL which is XHTML, CSS and Javascript. The engine underneath is the same (Gecko) but the frontend app over the top is what the user sees and controls buttons, menus, functionality.

Firefox was basically a fork of Mozilla stripped of the not-browser stuff and a cleaned up UI. It proved popular as a prototype so it grew into its own thing and Mozilla suite was abandoned. There is still a Seamonkey project that keeps Mozilla suite alive but it's outside of the Mozilla foundation.

The reason it's faster is that Mozilla was an entire suite expressed as a lot of XUL so it impacted loading times. XUL also had this neat trick that you could overlay XUL over the top of other XUL so the mail app was injecting buttons, menus and whatnot into the browser and vice versa. This was cached but it still had to be loaded. In addition and probably just as impactful, was that Mozilla shipped as dynamic libraries (DLLs) and a relatively small EXE, so it took time to start. In Firefox, the number of DLLs was reduced with static linking so it was more efficient to load.