In the west of Austria and east of Switzerland, the Silvretta constitues the border between both countries. The Silvretta is one of the most exciting areas in this region. The glaciers that make Silvretta famous, are mostly located on the Austrian side. A better contrast with the sunny green pastures is hard to think of. On the Swiss side, there are virtually no glaciers, and that's probably the reason that most visitors think the Silvretta is solely an Austrian matter.
Still, the highest peaks of the Silvretta are on the Swiss side. Piz Linard, 3411 m, is the highest. A second reason for exploring the Swiss side are the many old towns in this region, the Engadine. With some patience you might hear people speak Rhaeto-Romance. A multi-day hike through the Swiss Silvretta is a good option and perfectly possible.
However, as said, most hikers prefer the Austrian side. Armed with ropes, ax and crampons, they indulge in this high alpine landscape of ice and rock. By the way, if you plan to walk the Via Alpina, you will pass through the Silvretta.