Search me. Quite a few climbers are scared of heights too.
There's a huge gap between walkers who do a bit of scrambling or easy grade climbs and proper climbers.
I did a bit with some mates some years ago and you're right, proper climbers have a different way of dealing with fear.
For must people, it'd result in an adrenaline rush which gets you over the first few hurdles, but you can't keep on doing that. It's hard to explain, but they seem to replace that fear with something else which pushes fear and similar emotions way to the back of their brain. I think
@mrs the esk is a climber and might be able to explain it ?
Free climbers are a breed apart though. I've got huge admiration for all climbers, but free climbers are just amazing.