Some older frames have more relaxed geometry, which provides different ride and handling characteristics. It really depends on what you want the bike to behave like. I like the increased shock absorption and stability provided by slack angles and long stays, and don't mind forfeiting the lightning quick agility to get those qualities. Other people enjoy a bike that responds immediately to minor inputs. You can't have both so compromises must be made. Of course there are old bikes with aggressive geometry too.

Different tube sets will ride and respond differently as well.

Some people just enjoy the whole process of bringing an old bike back to life. Some of the C&V guys get a lot of pleasure from the challenge of the "Hunt" to find all the right bits and pieces to make the bike 100% period correct. I'd venture a guess that some of those guys enjoy building bikes as much as they do riding.

And, of course, many people just dig the aesthetics of old classics.