They can be the same in most cases. However, the expression give out can give out something that may or may not be physically existent while hand out can’t, since it requires a physical object. Examples: The officials are giving out relief goods to the needy. The officials are handing out relief goods to the needy. (Both are okay.) Our professor is giving out some extra points for early submission. (This is fine) Our professor is handing out some extra points for early submission. (This doesn’t sound right) Handing out something means giving something tangible or like an actual physical object to someone. Hope this helps 😊

In most contexts they are identical. You can also say, 'dole out', 'portion out', 'mete out', and 'toss out', although these expressions have additional meanings. "Handout" is also a noun, meaning the thing which is distributed. An action of 'handing out' normally implies it's a thing that's meant to be distributed, such as: "The teacher handed out the tests" "The application forms were handed out" An action of giving out, on the other hand, often implies a donation. "They gave out clothes to the needy" "They gave out advice for the uninitiated" "Give out these umbrellas to those soaking wet tourists" "Distribute" would be a synonym which would also work in both cases.

In British English they mean exactly the same, I think, and can both be used figuratively. You can also use "dishing out" to imply extravagant and rather careless giving of large amounts. However, "hand out", when used in terms of giving to poor people, can have a slightly negative connotation - perhaps as if the money is being given along with a little contempt for the poor people.