I'm not going to suggest that 'imply' is clearer/less clear, but they both mean what you think 'prove' does. To quote a bunch of dictionaries, the phrase is using 'imply' to mean:

'to involve something or make it necessary'

'to have as a necessary consequence'

'to involve something as a logical consequence'

Thus XKCD's extension of the phrase to `Correlation doesn't imply causation, but it does waggle its eyebrows suggestively and gesture furtively while mouthing 'look over there'.` -- which is essentially what you're getting at.