At least in singles skating?
Tara vs Michelle: Tara had her slew of 3-3s, Michelle had the artistry but this rivalry was over before Tara matured, who knows if she might've developed stronger artistry since she retired.
Mao vs Yuna: Mao has her 3A but she has also worked hard on her artistry and while Yuna doesn't do anything new in terms of jumps, she is hardly a slouch either. This seems like the most balanced rivalry with both skaters having good qualities in athleticism and artistry.
Tonya vs. Nancy: I don't really see Nancy as anything special in artistry but clearly over someone like Tonya who didn't exactly try very hard to be graceful.
Brian Orser vs Brian Boitano: Orser was seen as the more artsy skater but again, not like either were slouches in both departments.
Plushenko vs Yagudin: Yagudin definitely seen as the most artistic, Plushenko was the one pushing the tech but again the age difference and early retirement of Yagudin makes you wonder how this would've continued if they were still rivals in like 2004-05.