AROMA:
from Latin armata, pl. of arma, from Greek, aromatic herb.

FRAGRANCE:
from Latin fragrantia, from fragrare ‘smell sweet’

PERFUME:
(this *should have been a no-brainer....)
French parfum, from Old Italian parfumo, from parfumare, to fill with smoke (from Latin per-, per-) + fumare, to smoke (from Latin fmre, from fmus, smoke).

Perfume in the sense of 'smoking through' a scent, adds a whole new dimension to, 'perfuming one's hair'.

Alert: ramble on Indian hair perfuming ritual
At my grandmother's home in Southern India, this was a biweekly ritual. Hair would first be oiled intensely with a mixture of hair oils (they would really rub it into the scalp and not be satisfied that it had penetrated, till the oil would come off their hands and not stick anymore on the scalp); then we would be sent off to sit in the sun in the inner open courtyard. After sunning our oiled tresses like this for about twenty minutes, we'd have to wash it all off with a specially formulated herbal powder that did a great job of removing the grease. The next step was drying the hair... and this is where the perfuming bit comes in. A lady would arrive with a two chambered censer ; the bottom one for the hot coals (the perfumed incense is sprinkled on top of the coals) and the perforated top through which the perfumed smoke would exit. This censer was passed repeatedly through the wet hair, drying and perfuming it at the same time!