I'm from Independence, MO (suburb of KC), and I am not a fan of St. Louis, I know it has a nice children's science museum and the arch... but that's about it. So I will argue for Kansas City, since its my home town
Nelson-Atkins Museum of Art, one of the most overlooked art museum in the country, VERY nice.
Country Club Plaza, down the street from the Nelson, the country's FIRST shopping center, the architecture is modeled after Seville, Spain.
Liberty Memorial, officially designated as the national WWI memorial
Union Station and Science City (one and the same), across the street from Liberty Memorial, and Crown Center (think Hallmark Cards), beautifully restored about 20 years ago, amazing plaster ceilings, nice kid's science museum, planetarium, IMAX, and model train museum, it also has rotating traveling exhibits.
KC Royals, until last year bottom feeders of the MLB organization, not so much last year. Kauffmann was recently renovated, and is considered one of the nicest baseball stadiums in the country.
Kansas City Zoo, St Louis used to be better, but I think after all the expansions, and refurbs over the years KC is now far superior, just recently added a nice Penguin exhibit, and has a LARGE but very nice African exhibit.
Truman Library, home and museum, in nearby Independence (down the street from me!), beautiful library and home of US president Harry S Truman. Also in Independence is the Waggoner Estate, which is a beautiful old southern-style home to visit and tour. If you do visit Independence Clinton's Soda Fountain is still in operation, this was Harry S Truman's first job.
8th and Vine, Negro League Baseball Hall of Fame, I sadly have never been, but heard its a very nice tribute to the old Negro baseball leagues, the KC Monarchs was one of the best teams.
Freight House, home to one of the area's Jack Stack BBQ (YUM!) establishments, and also an up incoming eclectic art district. Located right behind Union Station.
Worlds of Fun/Oceans of Fun and/or Schlitterbahn. WOF/OOF are located just north of the city, typical but fun amusement park, paired with a water park, VERY nice waterpark, one of the oldest in the country, and one of the largest land-wise, they were built as separate parks and combined into one ticket two years ago. Schlitterbahn is much more new, located in KC, KS, claim to fame is the world's tallest waterslide, Verruckt (it means "Insane" in German, very appropriate). There is a Great Wolf Lodge out there too, shopping area and Kansas Speedway.
Oh yes.. you cannot come to Kansas City without trying the country's best barbeque! Its legendary.
Jennifer