They are the same : Dual Clutch, Direct Shift - same kind of box - just VAG and BMW-speak different. They are BOTH dual clutch computer-controlled manual gearboxes. VAG (Audi) uses the foot-off-brake to engage the first clutch while BMW uses foot-on-throttle to do the same. So one (VAG) acts more like an automatic (torque-convertor) and the other will roll (or not) with foot off the brake (and NOT on the gas) like a manual...Its not DSG BTW.
DSG is a true dual clutch manual. Audi will be using the S-tronic dual-clutch automatic and will be trying to replace their 8 speed auto with it.
S-tronic is a DSG (Direkt Schalt Getriebe) - it has two clutches, and VAG was first to production with it. BMW's earlier SSG (Sequentielle Schalt Getriebe, anglacized to SMG - Sequentielle Manuelle Getriebe - later changed to Sequentiellen M Getriebe) was a single clutch unit (ala R-tronic) but everybody has now decided that 2 clutches are the way to go - especially for launch control. BMW just called theirs a DCT instead of a DSG - semantics and a few hydraulic and control differences - BOTH dual clutch computer-controlled manual gearboxes.