"To hand out" and "to give out" mean the same thing.
"Are you going to hand out Halloween candy?"
"Are you going to give out Halloween candy?"
"Hand-out" can also be used as a noun, such as in the phrase, "I got these hand-outs at the presentation."
In English, we say "deliver" when talking about things such as mail, messages, etc.
"My parcel was delivered today."
"What time do they deliver the mail?"