In the first set of two sentences, the first sentence has only two people; let's say Tom and Jim. (Tom doesn't owe Jim anything, and Jim doesn't owe Tom anything, either.) However, the second sentence has a third person (him). The word "him" refers to a third person, Bill. For example, Tom is talking to Jim and says, "I don't owe Bill anything, and neither do you." Neither Tom nor Jim owe Bill anything.
The second set of 2 sentences have the same meaning.