They affect the "flavour" of the trip.
Yellow is generally accepted as the best to use in an ayahuasca innitiation. My experience is that it's never been intense enough for me; the trip is usually pretty mellow and easy to handle with pretty mild visuals and more of an acid/mushroom type feel. I have a pretty large quantity of this stuff laying around so I like to drink some by itself a day or two prior to a real trip.
White vine is what a shaman would give to someone who needs healing. If you have a disease or mental condition that needs to be assessed, this is what you'd take. The effects are generally similar to yellow vine for me.
Black is some scary shit. It is used only by the most experienced of shamans to make contact with the dead. It causes lots of very scary, life-like hallucinations of snakes, insects, blood and other menacing shit, accompanied by lots of shaking. I've used black vine once and I think it scared my sober wife more than it did me because I was having full body convulsions for a good two hours. There was nothing I could do to stop shaking. It also causes the most violent and extended purge. I would never touch this stuff again without an experienced shaman present.
Red vine is what is most commonly used by a shaman to facilitate the healing of others. This is pretty much all I use because it produces the best trip for me. It's also the most similar to smoking DMT (in my very limited experience). The visuals and hallucinations become indistinguishable from reality but rarely do they become evil or frightening. I usually feel like I become a bird or a jaguar for most of the trip and all five senses are affected. Almost always is a beautiful experience with a long-lasting afterglow.
All of them are brown in appearance and look similar but when you boil them; red vine will turn the water red, black vine will turn the water black/ dark purple. White and yellow will both turn the water yellow/orange/grey.
Hope this helps.