The word attar is used loosely enough now to mean almost any oil-based fragrance. Traditionally it meant something far more specific, and the specific meaning is the interesting one: a distillate received into sandalwood oil rather than collected as a separate essence.
The difference is the receiver
In conventional perfumery, a botanical is distilled, the essential oil is separated from the water, and that oil is later blended into alcohol. In the Indian method the vapour is led instead into a receiving vessel that already holds a base of sandalwood oil. The aromatic compounds condense directly into that base and are absorbed by it as the distillation runs.
The result is not an essential oil and it is not a perfume in the Western sense. It is a single material in which the flower and the sandalwood have been married during distillation rather than mixed afterwards — and the two behave differently on skin as a consequence.
No alcohol, and what follows from that
An attar is worn neat, in tiny quantity, usually from a glass rod or a fingertip rather than a spray. Without alcohol there is no propellant and no sharp opening: an attar does not announce itself across a room. It starts quiet, warms with the skin over ten or fifteen minutes, and stays close.
It also lasts a very long time. Alcohol evaporates and takes the lighter molecules with it; oil does not. And attars keep well — many improve over years in the way a spirit does, where an alcoholic perfume is generally at its best within a few years of being made.
One is a composition you put on. The other is a material that finishes composing itself on your skin.
Mitti, and the attar with no Western equivalent
The most singular thing the tradition produces is not a flower at all. Baked clay from the river plain is loaded into the still and distilled into sandalwood, and what comes out is the smell of the first rain landing on hot dry ground. It is called mitti attar, and there is no equivalent in the European repertoire, because no European perfumery tradition ever thought to distil the ground itself.
A.001 is built on it.
What to look for, and what to distrust
Because the word is unregulated, a great deal sold as attar is fragrance oil in a jojoba or paraffin base with no distillation in its history at all. That is not a scandal — it is often a pleasant product — but it is a different thing, and it should cost a different amount.
Two honest questions separate them. What is the base? A real attar is received into sandalwood, and sandalwood is expensive, so the price follows. And where was it distilled? A seller who can name the town and the season is describing something that happened; a seller who cannot, usually is not.