I think they have a strong relationship. While I get most of my casual jackets with no padding, I made to sure to get some padding for my stroller and tuxedo.
I saw a rather famous menswear guy wear an unpadded (or very lightly) tuxedo, and thought it looked pretty bad.
I think we all agree that completely unstructured looks casual, so let's ignore that for a moment. Where is the relationship between varying degrees of structure and formality beyond perhaps you having a preference for more structure with more formal things? I make this distinction in response to your comment about thinking lightly padded tuxedos look bad. I personally have seen some very lightly structured tuxedos that I thought looked great. Does it simply come down to personal preference or can you justify your idea with some kind of explanation?