I think "standoffish" implies a deliberate attitude while aloofness might just be a part of a person's nature. "Standoffish" also has a negative connotation that "aloof" doesn't have in the same way, to me.
I agree with Andrew; not a commonly seen word. I think I ran into "aloof" when studying what breed of dog to get.
The American Kennel Association gives this as a partial description of the character of a German Shepherd Dog:
Temperament: The breed has a distinct personality marked by direct and fearless, but not hostile, expression, self-confidence and a certain aloofness that does not lend itself to immediate and indiscriminate friendships.
(Emphasis added by me.)
I'm fairly certain I did not ever use the word before studying the breed standard excerpted above.
"Aloof" is a more versatile word. You can have aloofness, but standoffishness sound really odd.