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WARNING - CATHOLIC ANSWER
A religious sister at the end of the Faith Formation class I teach at was talking about this difference last night.
Angels are pure spirit, like God.
"Angel is their office, Spirit is their nature" - St. Augustine (see CCC)
Angels do not need to be saved by Jesus.
They made their choice to serve God or not before time began and do not have a fallen nature brought about by original sin.
Angels cannot consume the Eucharist.
Some theologians believe that this, coupled with the vision that Jesus would come to redeem mankind and be born of a woman, is the reason for Lucifer's revolt.
The internet does not seem to be a good place to source this info, or my google-fu is a little off. Most of what we posit about the nature of angels is borne of private revelation.
So take it or leave it, I'll try to improve this answer when I come across the places I heard this information.