For Ohio, I would say the Over the Rhine neighborhood of Cincinnati is the most architecturally and historically unique neighborhood in the state. There just aren't many places in the country that look like this anymore, unfortunately:
https://www.google.com/maps/@39.1107...7i16384!8i8192
https://www.google.com/maps/@39.1087...7i16384!8i8192
https://www.google.com/maps/@39.1095...7i16384!8i8192
Columbus' German Village might also be considered one of the more unique built environments in the state, but it's very quaint and sleepy compared to OTR's intensity and urbanity.
In terms of culturally unique places in the state, I think Yellow Springs might be a contender. It's home to Antioch College, and has been a very progressive town essentially since its founding. It was founded in 1825 as a utopian community, was an important stop on the underground railroad, and banned discrimination based on sexual orientation way back in the 1970s when basically no one in Ohio (or the country outside of SF and NYC) was thinking about protecting the LGBT community. It's unabashedly 'crunchy' and home to many artists and creatives, including Dave Chappelle. It has amazing parks in and around the town, including Glen Helen Park which contains the actual yellow spring for which the town is named:
It's surrounded by deep red rural areas, which make it even more of an oasis of sorts. Little bright blue dot in a see of red. It's a pretty special town, I think.