If the character is in school, it’s college. They could be living on campus or still living at home and commuting. If they live on campus, this may be their first time having to do laundry and light housekeeping. Most likely, they are in debt with student loans, so your character is broke, unless they are a trust fund baby or their parents are wealthy.
For some, it’s a level of independence and responsibility that is foreign to them.
New adult books are about new experiences and freedom. They can decide to join a fraternity, blow off classes or ignore schoolwork. No one is going to call their parents or ask for an absence note. Maybe they’re introduced to hard drugs for the first time. The main character could have their first love or maybe it’s their second serious relationship.
If the main character is not in college or trade school, they may have their first full time job or working the gig economy and probably still living at home or have several roommates.
The issues are more adult like, more drinking, more drug abuse, more sex and more discussing sex. The years spent in college are a time of exploration, they have responsibilities, but are not completely liberated. They have one foot in adolescence and one foot in adulting. It’s an exciting time for self-discovery; they feel like an adult without a lot of the complications.
Adult books have adult protagonist that deal with serious life issues (marriage, divorce, domestic violence, child rearing, bankruptcies, crime, prison, serious drug addiction, extramarital affairs, etc.) and more explicit sex. The adult protagonist is more confident in who they are.
Depending on the genre, the books may end with the answer: the protagonist solved the crime or mystery, or the protagonist made a life changing decision, or the ending expounds on the book’s theme or conflicts.
Bonus link for more explanations and examples: http://blogs.slj.com/teacozy/2012/12/28/what-is-new-adult/