They are similar in meanings with subtle differences in usages.
외모 - this is a typical word used to describe 'the physical look of a person' often used to contrast with 'the inside of the person'
외양 - this is interchangeable with 외모, although 외모 is used more often than this word
외관 - this is often used for "the look of an object", emphasizing the outside of the object compared with the inside.
모양 - this is a very generic term for 'shape'. There is a Korean colloquium "모양내다" which means "to adorn oneself."