Among the more “inside” stories, we’ve got the epic St. Petersburg vs. Moscow Linguistic School feud, going back to the early 20th century. The most important (just joking, simply the best-known one) part is the number of vowels. They in Moscow say there are five vowels in Russian, but we know there are six of them!
(The vowel of discord is “ы”, the one usually transliterated as “y” and sounding like the Turkish “ı”. The letter for it exists everywhere all right, but the sound itself is believed by those of the Moscow Linguistic School to be a variation of “i”).
Seriously, practically any time phonologists from the two capitals meet in the same room, expect ы-related banter.