Some benefits of bone handles are that they lend themselves to a wide variety of colors and textures. They can be dyed almost any color and the different types of jigging are only limited by the imagination. Bone is also fairly inexpensive when compared to other natural materials such as stag, ivory, shell, or highly figured wood. The downsides are possible chipping or cracking from being dropped. Also, the jigging can wear down from extended carry and use. While many people love the look of well worn bone, I imagine there are also others who dislike it.
Horn is a bit different since there are a few types commonly used. When it comes to ram horn, the exterior can be used to have a wildly textured appearance like stag. The interior part can be used to have a nice smooth finish. Buffalo horn looks fantastic, but can have problems with shrinking and is also subject to attack from insects. Bull horn has a wild variety of color hues which doesn't appeal to everyone, especially if the covers don't match on both sides of a knife.
When it comes to durability I don't think either bone or horn can match wood or even stag. That said, if properly fitted to the knife and cared for, any of them should last as long as the knife does.