When are you making the trip? Columbus is a lot more fun on a football weekend in the fall, although lodging then is more difficult and expensive! Cleveland's Warehouse District downtown becomes more enchanting as the weather becomes warmer and sidewalk seating becomes available. The Cuyahoga River and Lake Erie also become much more enticing in warmer weather.
There probably isn't much of a cost difference between Cleveland and Columbus, depending upon the comparable level of accommodations chosen, etc.
Cleveland and Columbus are very different cities, as Cleveland remains an international harbor with some of its industrial legacy still obvious, especially in its cultural institutions as Cleveland was the world's richest city on a per capita basis just over a century ago, when it was the stomping grounds for John D. Rockefeller and other American industrial magnates.
Columbus, Ohio's capital, is a more of a government, education and services center. It has prospered as such, but also it has emerged as a national retail center as the headquarters city of the Limited and its affiliates such as Abercrombie & Fitch and Victoria's Secret.
Both cities have vibrant entertainment districts, but Cleveland is gaining some national recognition as a "foodie" center. Although Columbus is larger than Cleveland, Greater Cleveland's population is much greater than that of Greater Columbus. They are different cities and personally I would hate to say which would be nicer for a visitor interested primarily in each city's night life for a single night, as that experience might be more a matter of luck than anything else.
Why not check out both cities, stopping at one on the way down and the other on the way back?
Cleveland's Warehouse District and Flats are its leading entertainment districts. Columbus' leading downtown entertainment district these days is the Short North, on High Street between Ohio State and downtown Columbus. While school is in session, Columbus entertainment districts are swelled by college students.
http://www.cleveland.com/warehouseguide/
http://www.nauticaflats.com/venues/
http://www.clevelandbarhopper.com/Detail.asp?Id=7&Name=velvet%20dog
http://www.shortnorth.org/index.asp
Hotels located within convenient walking distance of Cleveland's Warehouse District include the Marriott Downtown at Key Center, the Renaissance (Cleveland's most historic hotel) and the Ritz-Carlton. Both Renaissance and the Ritz-Carlton are located in Tower City, an architectural landmark in the Midwest. You have to drive to get to the Flats, and it can be confusing for visitors to the city.
More reasonably priced accommodations are available at the Doubletree Hotel Cleveland Downtown/Lakeside, which is within walking distance of the Rock & Roll Hall of Fame and Museum and offers free shuttle service to other locations in the city.
In Columbus, parking can be difficult on the Short North, but High Street has excellent bus service.
Hotels within walking distance of the Short North include the Blackwell Hotel on the Ohio campus.
Rates under $100 for the Blackwell sometimes are available on Expedia, but the hotel offers only valet parking.
Also convenient to the Short North is the Hampton Inn and Suites in Downtown Columbus. It's located on N. High Street in the Arena District, right around the corner from the North Market (an interesting place to visit for Jeni's Ice Cream, if nothing else; Jeni's also has a store in the Short North).
Check parking fees at all hotels.
If you want to drive to the Short North and save some money on lodging, you could try the Holiday Inn Express on East Hudson, not far from Ohio State and the Short North, which offers free parking. Consider parking at the Wexner Center on High Street if you want to drive to the Short North.
COTA mass transportation:
http://www.cota.com/
An interesting restaurant in the Short North is the Northstar Café – Short North on N. High Street.
http://thenorthstarcafe.com/
The premier shopping district in Columbus is Easton Town Center and it's quite impressive, and it also offers entertainment and lodging options.
http://eastontowncenter.com/
Cleveland's comparable retail center (Legacy Village and Beachwood Place) is located about 30 minutes southeast of downtown.
Columbus is minor league IMO compared with Cleveland (literally, the Indians' AAA minor league baseball team now plays in Columbus) if other attractions besides entertainment are considered.
Cleveland features one of the five best art museums in the U.S. and one of the world's best orchestras. Admission to the Cleveland Museum of Art is free. If you can arrange to spend a day or even a few hours in Cleveland, the art museum would merit a visit, especially if you have any interest in the arts.
http://www.clevelandorchestra.com/html/index.asp
A short walk from the art museum is the Cleveland Botanical Gardens. Its outside gardens are beautiful in the late spring and summer. It also features the Glass House, which recreate Costa Rican rain forest (complete with butterflies) and Madagascar desert habitats.
http://www.cbgarden.org/Visit/Glasshouse.html
Cleveland offers the second largest theater complex in the U.S. after New York's Lincoln Center:
http://www.playhousesquare.com/Events/EventsViewAll.aspx
You can learn much about Cleveland and its entertainment options by reading the answers section above or by searching Yahoo answers for "Cleveland warehouse district," "Cleveland night clubs," "Cleveland to do," etc. Ditto for Columbus.