While they do both often express a reticent emotional state, the cause of that state is different: for 'weary', it is exhaustion from past events, while for 'wa

While they do both often express a reticent emotional state, the cause of that state is different: for 'weary', it is exhaustion from past events, while for 'wary', the concern is to avoid undesirable events in the future. There is arguably some overlap where someone is being wary because they are weary of similar situations turning out badly all too often in the past.