One can define a height with respect to a special metric so that some particular useful equation involving the height holds exactly, instead of approximately as for the Weil height.
For example, one can define the canonical height using a metrized line bundle. This puts it on the same footing as the Weil height and not defined only as a limit of Weil heights composed with various maps.
More substantially, the Faltings height was defined this way, using a metrized line bundle on the moduli stack of abelian varieties. The invariance of the Faltings height under isogenies associated to p-divisible groups was crucial in Faltings' proof of the Mordell and Shafarevich conjectures.
Edit: Probably the answer to the second question is one doesn't need the full machinery of arithmetic intersection theory, i.e. one can define the height associated to a metrized line bundle without invoking all the techniques needed to handle arithmetic intersection theory in full generality. One just needs to take a section and add up its valuation at each place.