Without specifics of what you like, it really can't be answered.
Art Museum : Slight nod to Cinci. But near the Cleveland one is an Ohio historical museum which has an incredible old car collection.
Science Museum : Great Lakes Science center is great. OTOH, the old Railroad Station complex in Cinci currently has an amazing Pirate exhibit which is worth a special trip.
Zoo - Probably Cinci.
Parks : King's Island vs Geauga Lake. Haven't been to latter in eons, so can't say.
Unique Attraction : Toss up between Newport Aquarium and RnR HoF.
Restaurants : Cinci has a pretty active scene right now with some more unique places like Pho Paris and other places spread out. Cleveland has some good places, but less singular.
Restaurants II : Cinci Chili, Goetta vs Pierogies/ ?? Gotta go w/ Pierogies. Skyline is only a notch below non-chains like Blue Ash, so no need to go there.
Beer : Great Lakes Brewery blows away anything in Cinci.
Nightclubs : Cleveland wins easily.
Arts and Shopping : I found more boutique shopping in Cinci though all nestled around in various places. Cleveland is pretty chain-y and I found the more boutique places were little different than ones in Columbus. Not been too impressed w/ galleries in Cleveland.
Shopping II : Cinci has Jungle Jim's and Party Source. 'nuff said.
Markets : West Side Market is pretty sweet, but precious. Some good stuff at it. Cinci's downtown old market (forgot name and location) is more urban, though has some good local farmers in season.
Theatre : A bit more active in Cinci, I think. Seems like more big stuff comes through. Not real sure on this.
Sports : Not gonna touch this one. No NBA in Cinci. Better college b-ball though.
Surrounds : Cleveland easily. KY vs Lake Erie. Though Covington/Newport/Bellevue is pretty nice in the old parts.
A.