What is the difference between 翼 and 翅膀?
翅膀 is a compound word only refers to actual wing attached to living creatures. Airplane's wing can only be called 機翼

翼 is a general term for wing. Beside describing all kind of wings. It can also be used metaphorically.  For example: '左翼' (left wing) '右翼' (right wing) in politic; '側翼'  (side wing) in military formations;  翼鋒 (left or right side striker) in soccer; 東翼 (East wing), 南翼(South wing), 西翼(West wing), 北翼(North wing) of a building

You cannot replace 翼 with 翅膀 in these examples