they're both cool for different reasons
chicago is cleaner thanks to its alley system and wider streets, it has a more gothic theme, the river, the el circling the buildings downtown which is unique, better parks and green spaces, better organized
ny downtown ( or i guess lower manhattan) has a lot; its gritty, dirty, its colorful , very international, eclectic, chaotic, crazy etc etc,
they're both pretty busy and crowded esp in the summer(ny is a bit more crowded) but you can tell the city of chicago was much better planned than Manhattan and ny in general.
chicago has better views of its skyline from a distance, in ny the only good view is from brooklyn.
But i'd rather live in DT Manhattan since DT chicago is mostly offices and skyscrapers. Lower manhattan actually has residential blocks.
If you want a true urban living experience in chicago you gotta go a little bit outside of downtown.