As Henry mentioned, the 35-34 game in 1979 that (combined with a lopsided victory by the Bears over the Cardinals earlier that day) kept us from winning the division and out of the playoffs likely remains the most devastating Skins loss of all time to longtime fans.
I'd have to say the 1984 divisional loss to the Bears was disappointing in a certain way. Granted, Chicago was an up-and-coming team and would steamroll the league the next season. However, that was the only playoff loss we ever suffered at RFK and that it meant that we wouldn't be going back to the Super Bowl that season after two straight appearances was a bit depressing at the time.
The 17-0 loss to the Giants in the NFCCG following the 1986 season was likely less so because they had beaten us twice already and the game was at their place.