Your understanding is correct. "Unless" cannot be used to replace "because ... not". It is logically equivalent to "if ... not". Your second statement means:
If you do not help me, I will not finish my work in time.
In other words, "unless" refers to conditional or hypothetical situations, not actual ones. When you're stating an actual fact—the reason X happens is that Y doesn't happen—you cannot use "unless".
The Education First site has a fuller list of examples showing how "unless" is the equivalent of "if ... not". Off the top of my head, I can't think of any equivalent one-word replacement for "because ... not".