It sounds completely trite but mostly to push my limits and to experience and see places most people never will. I've done every climb with my sons so there is a huge element of creating collective memories as well.
Beyond that it's hard to describe the sense of accomplishment when you stand on top of a mountain. I'm not sure how to put it in words. You get a real sense for the immensity of the world and just how tiny you are. It's humbling and exhilarating. I realize that sounds so corny but I don't know how else to describe it. I've said it before, everyone should climb a mountain.
As to the fear of heights, the first time I strapped into a fix line as we maneuvered on a ledge with a 50 foot drop I was scared to death. I just shuffled my feet along and took forever. I've been back to that same ledge about 3 times since and I don't even use the rope anymore, so I'm getting better. Also, a lot of hiking/scrambling/climbing is the drudgery of staring down at your feet as you "left foot-right foot-left foot-right foot" along a trail, interspersed with moments of thrilling exposure or adrenaline rush. It's not 12 hours of terror.