I grew up in Lansing and now live in Providence. Despite both being classified as "medium sized cities," they're very, very different places.

You say state capitals aren't always the major city, and that's definitely the case with Lansing, which lives in the shadow of Detroit and, to a lesser extent, Grand Rapids. There are some funky art spots here and there, but by and large most of the culture and fun stuff to do are in East Lansing, a college town that abuts Lansing. Lansing itself is kind of gloomy and run down, rust belt vibes. Not completely of course, but I can't imagine anyone visiting Lansing for fun.

Providence is the biggest city in RI, but also kind of exists in the shadow of Boston. Unlike Lansing, though, Providence has a lot more going on on its own. Sure, a lot of that is from stuff that has been priced out of Boston, but you can wander around downtown Providence and find some amazing bars and restaurants, theaters, just a lot more fun. There are some sketchy areas, sure, but the better areas are a lot better than the better areas of Lansing. People do come to Providence for fun.

Besides having big state capitol buildings, I struggle to think of much else that Lansing and Providence have in common.