The question that is the title of this thread has occasionally crossed my mind, just out of curiosity. Why aren't they Army, because they look like 'em and run around with guns etc?!?
It's easily answered:
1. Peacetime. An irreplacable link at the Stn level between those in the more blunt occupations and the demands of the *ahem* sharper end of warfare. You could, in theory, have Army rotated through that role, but that would not be popular or successful - not guaranteed of getting posts filled, culture clash, lack of understanding. etc The operational demands faced by all personnel have increased enormously since the Cold War and RAF Regt fulfils an invaluable training role.
2. Wartime. Offensive and defensive ground operations in support of RAF operations, described in more detail by those who know far more than I do, based around safeguarding an operating base. A RAF detachment wouldn't *really* want to have a random Army rifle company foisted upon it.
There's all the banter as well, but respect tends to be mutually earned and a little bit of interest and respect for someone else's role ensures a happy relationship outside the Respirator Testing Facility.
Long may the RAF Regt continue. Not quite as famous as the Royal Marines, but a only few hundred more years to go. Per Ardua, indeed!