Overlook effectively means something that you did not think about, potentially forgot, or generally failed to pay attention to. Neglect would be more along the lines of something that you purposefully ignored, though there are exceptions. I'd say they're interchangeable, but neglect has a heavier connotation to it.

In addition to Gabriel's fine answer, I would like to add that, as in so many cases, context is important here. For example, if we're talking about mistreated children or animals, then you would say that they were the victims of neglect. If you forget to pay a bill, then overlook is probably the better of the two. Or put another way, being accused of overlooking something might result in a facepalm or rolling of the eyes, but the consequences of neglect are usually more serious.