Rich:
Brettsomers has it right - Crawford's Best Actress Oscar was for a combination of things. It was part longevity award (Crawford made her first movie in 1925), part acting award (Mildred Pierce offered Joan a lot more to work with than the Depression-era shopgirl roles she'd specialised in at M-G-M), and part whip-cracking over Bette Davis. A big part of the reason Jack Warner had signed Crawford in the first place was as a "threat" to Bette, who had become a monumental pain in the keester. On Mr. Skeffington (1944), Davis had misbehaved and held up shooting so much that the movie's producers, Philip and Julius Epstein, had walked off their own picture - a situation I do not find repeated anywhere in Hollywood history. Warner's idea was to hold the threat of Joan over Bette's head by giving Joan parts Bette wanted if Bette was in one of her difficult moods.
Although Warner does not ever seem to have actually given Joan a part originally intended for Bette, the threat of having another big female star on the Warner lot worked very well. Joan got the Best Actress Oscar for Mildred Pierce in '45, got excellent reviews and box-office in 1946's Humoresque (the best of all Joan's movies, IMHO, though the lady herself didn't like how it turned out), and another Best Actress nomination in '47 for Possessed (unrelated to a 1931 movie of the same title Joan had made with Clark Gable).
Bette never recovered from this triple whammy of Joan's, evidently losing some of her confidence. She began making stinkers (1948's Winter Meeting and the same year's June Bride among them), and in 1949, she got into such fights with director King Vidor on the set of Beyond the Forest that Warner called a summit meeting. Bette, wanting Vidor off the picture, told Warner: "It's him or it's me!" Weary of Bette's didoes, Warner didn't bat an eye: "Okay, Bette - it's you!" Shocked beyond measure, Davis quietly finished the picture and packed up her dressing room.
It's this situation that led to the epic feud between Crawford and Davis, which was much more Davis's doing than Crawford's - Joan actually admired Bette's accomplishments on-screen. But Bette was too humiliated ever to let it go, as Joan would find out to her cost years later on Hush, Hush Sweet Charlotte, where Bette played such mind games with Joan that Joan took to her bed (and some say the bottle). Bette got Joan replaced with Davis's good friend Olivia de Havilland; the scenes Crawford had finished were re-shot.
It didn't help that Bette and Joan had been offered a choice of straight salary or a percentage for Whatever Happened to Baby Jane? - Bette took the salary ($350,000, if I'm correctly informed), and Joan took the percentage. Baby Jane was a monster hit (no pun intended!) and Joan ended up with around $2 million for her work on the film - at a time when Elizabeth Taylor's $1 million salary for Cleopatra (1962) was headline news.
Anyway, Joan was a great star, and Bette was a great star, and it's a damn shame that studio politics set them at such odds. Their pairing in Baby Jane was dynamite, and it would have been nice to see them work together in several hit movies, showing Hollywood that you didn't have to be twenty-one to pack a theatre - a lesson Tinseltown has yet to learn.