you are allowed to cover almost anyones songs without the songwriters permision if you do a "straight" cover version, but if you want to change more than the odd lyric, add lyrics or change the arrangement, you need to obtain permission from the copywrite holder which is usually the songwriter
thus mariah could cover it without his permission. i presume beyonce's arrangement was different to a straight cover, possibly part of a medley, or lyric change due to gender change, and that's why it was refused
prince hates other people singing his songs. in particular what he really hates is people making bad covers and changing the arrangement of the song. he commented on ginuwines cover of when doves cry that timbaland produced, in that he changed lyrics without his permission and he didn't like the change
he also doesn't like people sampling his music either and will usually turn down sample clearance, although people like 2pac have sampled him in the past. he also usually denies anyone to use his songs in movies. what he doesn't mind so much is the use of the songs that he gave away to other artists. whilst prince has released over 30 albums under his own name, he has given away enough songs to fill another 30 albums for artists like the time, sheila e, vanity 6, appolonia 6, carmen electra, the family, madhouse, tamar, bria valente, mayte, elisa fiorillo, chaka khan, larry graham, etc etc. and thats just stuff where he's basically created the whole album/track for the artist to sing on. he's also given songs that have been released by kenny rogers, celine dionne, joe cocker, madonna, sheena easton, candy dulfer and patti labelle (and loads loads more)
the unusual thing about the beyonce case is they rehearsed together at his studio in minneapolis and performed together at the grammies in 2004. so it doesn't matter how close you are to him, his rules are rules. he isn't so close to mariah but he has mentioned discussing the music business with her
because prince writes, records and produces his own music he feels he should have complete control of it rather than someone else like a record company, and likewise other people shouldn't be allowed to use his songs without his permission. the problem with that is the laws that surround the music business that allow people to cover songs without permission. this harks back to his legal dispute wth warners in the mid 90s when the staff at the label who were around in the good old days when labels encouraged artists to create good music had left and been replaced by managers and accountants that knew nothing about music and were only interested in profits. that's why since 1996 prince has kept control of his music and only released on major labels as part of a distribution deal for the individual album. he still retains the rights to do what he wants with those albums, such as give them away with a newspaper or at a concert
his issue with youtube, and to some degree other websites, was that people were making money from his work, and he wasn't receiving a penny for it. thats why he instructed universal to block any of his work from youtube. he has now reached some deal with youtube so he will be paid if his work appears there. he has recently released a few videos on youtube in the form of viral marketing. another reason for concern with fan sites was that they could illegally post videos of prince works like tv appearances, but prince couldn't legally post the same performances on his official website without obtaining permission from the shows (and many of the american shows won't grant this permission for anyone for any reason) and without paying fees. thus fan sites could illegally host more than he could officially, reducing his capacity to generate a profit from his own work
he's since stopped trying to close the fan sites down, but he still has certain issues with the content on some of them, in particular the discussion of bootlegs. he plays live a lot, and his shows vary a lot, so there is high demand from fans to get a copy of every show he does, but prince doesn't release live albums or videos often, whilst almost every show is bootlegged. this is because live performances are the best income stream for him, and he prefers that people pay to attend live shows rather than listen to them at home on cd or dvd. this may sound strange to fans of most artists who would be more than happy with one or two soundboard quality recordings from each tour, but prince's catalogue is so vast and his band are so rehearsed that he constantly changes setlists, and even when he does play the same song it can change style dramatically from one day to the next. some artists will stick to playing 75% of the same setlist from night to night, and change a few songs, but they have to play all the hits, whereas prince will do shows without playing any of his hits at all, and he will do this frequently. he may play a large show full of hits and new songs, and then a second smaller show right afterwards that's full of obscure tracks, cover versions, and album favourites. this year he's been pulling out ancient tracks that he's never played live before, but really really great tracks, and he has some interesting arrangements for more familiar tracks, so some of the shows in the last 12 months have been some of his best ever, which is pretty unusual for someone who's been playing live every year for 30 years