"Apropos" means (ADVERB): 1 : at an opportune time; seasonably 2 : by way of interjection or further comment with regard to the present topic (ADJECTIVE): 1: being both relevant and opportune
"Appropriate" means (TRANSITIVE VERB) 1. to take exclusive possession of; annex 2. to set apart for or assign to a particular purpose or use 3. to take or make use of without any authority or right (ADJECTIVE) 1. especially suitable or compatible; fitting
As adjectives, they are nearly identical in definition and/or use. I think the primary difference lies in the "opportune" part of the definition of "apropos". Whereas some thing or action could be appropriate at any given time, when it is particularly appropriate because of particular timing or circumstance it is apropos.
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