Translation:On the contrary!
"kontraŭ" means literally just "against". A real opposite would be "malo" or in the -e form "male".
So I guess "kontraŭe" is being used in situations when you just want to express another opinion (not necessarily a direct opposite) on something. Maybe that's what "on the contrary" does in English, I don't know.
I'd never actually thought of that before, but I think you're right, "Male, ..." does sound a bit stronger than "Kontraŭe, ..." as a way of introducing a counter-argument, because "male" means "I'm about to say exactly the opposite of what you said," whereas "kontraŭe" is more like "On the other hand..."
Of course, you could say "On the other hand," and go on to flatly contradict what you've just heard too!
To me "male" just sounds like it means "the other way" - adverb of manner and not a conjunctive or interjective adverb.
But, there isn't a strict difference between the adverbs that add emotional/logical nuance to what you're saying and the regular ones that just describe verbs/adjectives. So, I suspect it just depends on usage.