I have been to both and spent considerable time in both.
We found a lot more jazz and blues in St Louis than KC. The St Louis downtown also seemed brighter more energetic clean and more vibrant. St Louis seemed a newer more shiny city, with lots of jazz and blues. The St Louis Jazz Society owns their own venue and produces shows 5-7 days a week with local and national acts.
IN KC I hung out Friday and Saturday nites (well, early mornings 1am-5am) at The Mutual Musicians' Foundation Building, where show started at midnight. But that was just before COVID. I don't know if they're still doing that. While in town, besides that historic venue, I could not find a lot of other jazz or blues. The jazz museum was underwhelming.
If I wanted to move and my 2 choices were KC or St Louis, I'd choose St Louis. If not for the music then for everything else about the city.