Hand out is to give - number things usually to a number of people. “The teacher handed out exam papers to all the students” Or a charitable gifting of something: “He was very poor but didn’t like to accept hand outs” Hand over also means to give something to someone but it’s usually a singular thing. He had to hand over his gun to the police. He had to hand over the keys to his house.

just a quick clarification: Jack used the phrase "hand outs" in the second sentence referring to the poor man not wanting to receive "hand outs" .... Be careful of this usage. In this case hand out becomes one word: HANDOUT and it becomes a noun. The phrase hand out vs hand over, you asked about is an verb or action. They are doing something because you can use the word HAND as a verb - "can you HAND me a napkin?" or "He must HAND over the money he owed" .

Hand out refers to anything that is given freely. "I will hand out free samples at the restaurant" Or "The teacher will hand out the assignment. Hand over refers to something that must given. "Her parents made her HAND OVER the keys to her car" Or "I had to HAND OVER the money that I owed"

Hand out my money, please. Hand over on her social studies class.