As far as I know, both 翼 and 翅膀 mean "wing". I've heard people say "雞翼", "雞翅膀", "固定翼飛機", "鳥翅膀". All of these are wings, so what is the difference between 翼 and 翅膀? When should one be used over the other?
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
翅膀 is Modern Vernacular Chinese, while 翼 is Classical Chinese.
Think of Classical Latin terms used in Modern English, such as "de facto", "de jure", and "etc". The difference between 翅膀 and 翼 is just like the difference between "fear of heights" and "acrophobia".
鸡翼 / 雞翼 = 鸡翅膀 / 雞翅膀 = chicken wing(s)
However, 翅膀 is preferred to be used for living animals, such as 鸡翅膀 (chicken wings), while 翼 is more often used for aircrafts or politics, such as 机翼 / 機翼 (aircraft wings) and 左翼 (left-wing politics).
I think that it is mainly because technical terms are always chosen from classical words while colloquial terms are mainly chosen from vernacular words. This phenomenon also exists in English: when talking about technical terms, people choose words from Latin or Greek, such as acrophobia, pedophobia, xenophobia; when talking about colloquial terms, people choose words from English, such as fear of heights, fear of children, fear of foreigners.
users recommend consulting online dictionaries noting large number of usage examples, bkrs: 翼I b.f.（bound form !) 1) wing (双翼) ①<名>鸟类或昆虫的翅膀。 2) flank (左翼) examples of bound form use: 右翼 左翼 两翼 侧翼 羽翼 机翼 不翼而飞 鼻翼 比翼 如虎添翼 极右翼 比翼齐飞 南翼 双翼 尾翼 比翼鸟 蝉翼 卵翼 水翼船
翅膀 unbound！(free)① 翅通称。 ② 物体上形状或作用像翅膀的部分：飞机翅膀。 某些动物用以飞行的器官或某些象翅膀的东西 鸟类及昆虫的翼CL:个,对 (naturally no CL for 翼)