Swensen's was a fairly popular restaurant chain in the 1980s. I'm not sure I ever went to the Columbia location (now The Hunter Gatherer) at the corner of Main & College Streets, but almost anywhere we went on a trip, there would be a Swensen's. I know for sure there was one on The Market in Charleston (now an Applebee's, I think) , and we ran into them on class trips to DC and Florida as well. The ad above from the 1985 Southern Bell phonebook has the logo I recall.
Swensen's started in San Francisco as an ice cream stand, but by the time it franchaised and locations hit the Southeast, they were casual dining restaurants (with ice cream, of course) and I think I had burgers there more often than anything frosty. Their fries were a bit unusual in that rather than being longer than they were wide, they were sort of square and waffle-hatched.
According to Wikipedia during the 1990s, the chain shrunk from 400 stores to about 200, and when it started to expand again, it was mostly overseas. I think the Columbia store closed during that wave of shrinkage. The current tenant in the building, The Hunter Gatherer brewpub has left the interior in a rather rough (if interesting) form. I suspect it was somewhat less distinctive as a Swensen's but I could be wrong. I would be interested if anyone can recall whether Swensen's had the main-floor and catwalk layout used by THG.
UPDATE 16 April 2010: Added Campus Club South and TW Muldoons to the post title and identified what year the ad is from. Added The Quarter Moon to title.