It has nothing to do with the height. Both rockets go 'up' a similar amount. The difference is how far sideways Falcon 9 has to go (since it is launching a payload to orbit).
They're in that ballpark. I know the capsule exceeded 100km. I don't know if the first stage technically broke that or not, but it was closer if it didn't.
I think SpaceX are in a much higher ballpark?
Not really... A few dozen km altitude difference doesn't mean a whole lot. The _much_ more significant difference is that Falcon 9 is traveling sideways at several times the speed of sound. New Shepherd is "just" going sight up and coming back down.