A diversity statement can have properties of an adversity statement and vice versa, but overall they should have a different big-picture and way you present your topics.
Diversity - What about you is cool and unique, what useful skill/talent/quality/wisdom do you bring to share with your classmates and will help you as a medical student? You can talk about culture but it will have to be about how it makes you different (in a good way) and how that "different" is beneficial to the community/peers/patients/school/etc.
Adversity - What challenges did you face in your life, how did it affect you, how you dealt with it, what you learned, and how you grew. The last three points are the most important; here you're focusing on how you overcame your challenge regarding culture and what you learned and how it shaped you.
Just a premed, take it with a grain of salt.