The trope pages seem quite distinct:
Odd Couple: Specifically limited to rigid outlook versus relaxed outlook, plus both are main characters. Riggs and Murtaugh from Lethal Weapon are an Odd Couple
Odd Friendship: any pair that would not be expected to get along, but who are close friends; may hold any place in the cast. Harold And Maude are an Odd Friendship but not an Odd Couple.
edited 9th Nov '10 10:30:51 AM by Madrugada
I'm not sure why "main character" is or should be a required feature of Odd Couple, but I agree that it's a quite distinct trope from Odd Friendship. The former is more of a conflict trope than a friendship trope. The Odd Couple don't necessarily have to be friends; they may just be thrust together because they're co-workers or inmates at the same prison or the like.
I would say that Azriphale/Crowley is definitely an example of Odd Friendship; calling it Odd Couple is a stretch at best. On the other hand, a fairly clear example of Odd Couple where the characters are neither main characters nor friends would be in Going Postal, where two postal workers share an office with a line of white paint down the middle, where one side is pristine, and the other side an unholy mess.
As for the name, the work is The Odd Couple, so there's no conflict, but if we want to avoid confusion with either the work or the Odd Friendship trope, I might suggest renaming Odd Couple to something like Roommate Mismatch. (Likewise, if Odd Couple is definitely only about the main characters, then we need a broader trope to cover the many examples that aren't, and something like Roommate Mismatch might work.)
Speaking words of fandom: let it squee, let it squee.

Courtesy links: Odd Friendship and Odd Couple
Based on the descriptions themselves, the only distinction between these two tropes is that Odd Couple is when the characters involved are the main protagonists, and Odd Friendship is when they're not.
Given how the trope pages are actually used, however, few people bother to make that distinction. There are examples listed on both pages (Kirk/Spock/McCoy, Superman/Batman, House/Wilson), examples clearly misfiled (Aziraphale and Crowley), and some borderline cases.
While it would be possible to clean up the examples, I think the distinction between "main vs. secondary characters" is too vague to be usable, especially in ensemble works. I'd recommend merging the two tropes into one, preferably as Odd Friendship (since "Odd Couple" is a title of an existing work).
Thoughts?
—R.J.