"Rely" weighs a little more than "depends". If you rely on somebody for something, you may not have the capacity to do it yourself. If you depend on someone, you may actually have the capacity to carry out but don't want to or have time to do it. I rely on my accountant to handle the annual tax form because he is more skilled at doing this type of work. The US has depended on foreign oil for decades.

They are ever-so-slightly different but I'm not quite sure how to explain why. "Depend" usually means the way in which something happens, whereas "rely" is if it happens at all.

yes. they are synonyms.