The second sentence is incorrect. You can not “have” something before it has become.
They now have an interest in snakes because they became interested in them in the past.
December 2, 2017
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There isn't much of a difference- but I would always use the first option, 'when did you become interested'. I don't think I've ever heard someone ask when someone else 'has become interested' in something.
December 2, 2017
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