I am not sure there is a standard definition for adventurer versus hero, but in modern terms I believe it would be a non-professional or weekend warrior, versus a professional or skilled athlete. The biggest problem with 5E is the classes are not treated in the same fashion starting at first level. So fighters would tend to be adventurers because what they do is not very defined or limited, and they have to wait longer to start to diversify or specialize where a bard, wizard, or cleric is a professional right at the start.