The first sentence should end with "he was giving in" and is written in past progressive tense, which is used to describe an action that was happening at that moment. Since he "was giving in," he may not have completely given in by the time all was said and done. The second sentence is past tense only and indicates that the action has concluded. "He gave in," so there's no question that he finished the action he was doing in the first sentence.

First one= "... was giving in", not "...gaving in". Also, he is in the process of giving in. The second one= "... gave in", he already given in. Probably the difference is just in the first sentence you can see the process of giving up, while the second shows he has reached the goal of giving up.