Tallness is the length of an object that it typically upright -- an absolute measure. Or it is a relative measure: comparison of such lengths.
Height is a distance "above" an origin position or surface.
Height applied to objects, in particular persons, can mean how tall they are: Tom's height is 6 ft 2 in. The height of the Eiffel Tower is _ .
But in a context where the position of the person or other object is meant, that is, it is not being measured itself but its distance from something else is being measured, height refers to the distance of that position above some base (e.g. sea level -- altitude).
IOW, the meaning of height depends on the context: whether you are measuring a person's body or the distance of the person from something. By default (with no further context), the former is meant.
Although height has these two meanings, "high" does not. Tom's height is 6'2", but he is not 6'2" high.
" Tom is higher than Sue " means that Tom's location is above Sue's. Tom's height on the mountain is greater than Sue's; he has climbed to a greater height.
" Sue is taller than Tom " means that the length of her body is greater than the length of his: Sue's height (6'3") is greater than Tom's (6'2").
" Sue is taller than Tom " means that the length of her body is greater than the length of his: Sue's height (6'3") is greater than Tom's (6'2").
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