Shivers usually means cold. Example: "I was shivering all morning because it was so cold." Chills can also mean you are cold or you have a sickness, but the expression "gives me the chills" will be understood as something that moved you to the point of having goosebumps. Shivers may sometimes be used to suggest something scary. Example: "We watched a horror movie this weekend and I shivered with fear." For the sentence you mention about liking a good song, I would say, "The song was so good it gave me the chills!" or "The song was so good it gave me goosebumps!"

These are idiomatic expressions, so their use may be different from one country to another, or they may not even be used in all English speaking countries. That being said, I'll give you my answer from an American English standpoint. You could say it either way and it would mean, essentially, the same thing. although "shivers" is less commonly used than "chills" in this context. As Sara mentioned, you could also say that something gave you "goose bumps". A less common, but valid alternative, is "goose pimples". So, you could say that "the song was so good that it gave me goose pimples". All of these mean the same thing and, in my opinion, most any native English speaker (at least from the U.S.) would clearly understand your meaning, regardless of which of these you chose to use.

I think both can be used to describe a sensation of shaking from cold or fear. In my opinion something "giving me the chills" will only mean a feeling of fear or dread. So like Sara said below, something beautiful might give me "goosebumps" not the chills.