Great thread and here is what i know of my own knowledge and what i looked up
Kate Bush did two Wuthering heights one
One is where she is outside in a red outfit and the other is in a dark room with bright lights and a white flowing outfit.
Mariah carey did a few alternate videos
For All i want for xmas, she did a second version but set in the 1960's
She also did alternative remix videos featuring rappers for songs such as Fantasy, Obssessed, Always be my baby, Honey etc
Britney did two for Till the world ends. The first is more storyline based with people rushing about preparing for the end of the world. The second cuts that out but includes more scenes of Britney dancing.
Kelis did two Milkshake videos. One is where she is outside and it involves an ice cream van. The second which is the best known one where she is in the diner. The original for some reason was shelved but can be seen on You tube.
Beyonce did two versions of Listen. One is where she is on stage and the other involves a photoshoot. Halo also has an alternative video. In the second video, it's really dramatic and involves her lover dying
Colbie caillat shot two videos for The Little things. The first one is shot in San francisco and shows Colbie showing interest towards a guy. In the second, it was filmed in Hawaii and involves her riding a bike and meeting the guy she likes at his work.
Jordin sparks did two for Tattoo. The first invovles her on a merry go round. The second involves different couples falling in love and have tattoos appear on their bodies.
Leona did two for Bleeding love. The original is her in a house and it shows different couples. The second is her walking around a city while other shots include different couples.
Florence did two for Dog days. The original is one of her done in the woods. The second is of her in a white room.