These two words are both adjectives and they appear to have somewhat related meanings, but they are not at all interchangeable. Yes, they both have a negative connotation to them, but the definitions are not identical. This situation is similar to imply and infer, for example.
Adverse is an adjective that means preventing success or progress. It means to be antagonistic and unfavorable.
“The ship encountered adverse weather conditions.
Without being specific, the above sentence implies that the seafaring vessel may have come across harsh winds or a strong storm. These weather conditions are not conducive to the ship getting to its destination.
Averse, on the other hand, would mean to have a certain repugnance or an opposing feeling toward something. If you don’t like something and you’d rather avoid it, it may be appropriate to say that you are averse to that something.
Tim is averse to flying, because he doesn’t like how taking off makes his ears pop.
This sentence is saying that Tim dislikes or is against flying in an airplane. If you were to say that Tim is adverse to flying, you would then be saying that Tim is somehow preventing, impeding, or otherwise obstructing planes from traveling.
Adverse grammatical rules and exceptions can make many people averse to learning proper grammar in the first place. Hopefully this series of posts on Beyond the Rhetoric can clear up some of those confusions.