The west side, running down the Gerlache Strait, is the classic Antarctica people picture: huge glaciated mountains, whales, colonies, with a stop in the South Shetlands on the way. Most classic and extended trips go there. The east side, meaning the Weddell Sea and the Antarctic Sound, is a different landscape, less of the big mountain look and more tabular ice and open water, and it usually shows up on South Georgia itineraries because it sits logistically closer.
The east side also gets sold on its own as a Weddell Sea trip, generally aimed at repeat visitors who have already done the classic route.
Two things worth knowing. Early season east side trips carry a chance of emperor penguins, and a chance is all it is. Never book on the expectation of seeing them.
And weather can reroute a west side trip up and around into the Weddell without you having chosen it. That is the expedition team finding you a better day.