There is a subtle difference, although in the examples you cite, there is really no difference in meaning:
The old man gave all his money to the people in need.
The old man gave away all his money to the people in need.
Usually "give away" means that you give someone something that is yours - for example, you would not usually say this about a gift that you had specifically bought someone, because you never intended to keep it for yourself. In your examples, both statements make it clear that the old man's money was his, so there is no difference in meaning.
Also, "give away" tends to imply that you the item(s) have gone far from your sight. I would not use "give away" if, for example, I gave a possession of mine to my children, because the item is still within my family. I would just that "I gave it to my child"; if I gave the item to someone beyond my family circle then I would say that "I gave it away".