As adjectives, they are synonymous and sometimes interchangeable. You would have to see the words in context to collocate them naturally. "external" and "internal" are often used when referring to a surface e.g. a wall, a layer, the surface of a bodily organ, and you want to distinguish one side of the surface from the other. "exterior" and "interior" are often used when referring to a more general area either side of a boundary or surface. e.g. interior decoration takes place inside a house, and exterior decoration takes place outside a house. "exterior" and "interior" are also used as nouns to refer to areas ("the exterior of....")

Exterior - situated outside; the outside part of something 'The exterior of the house needs painting.' External - of, on, for, or coming from the outside 'The external walls of the house need to be painted.'

External refers to outside. Exterior refers to the outside of something.