I see this issue all the time and it is about theme and focus for each essay. You may have similar content but framing it for the specific question matters. For example, using the previously LGBT, the "Challenge/Adversity" essay could be how you told parents, dealing with issue that may have arisen in schools, etc. The "Uniqueness/Diversity" essay could be about what you could bring to your patients and relationships with them as a being part of an often discriminated or stigmatized group.
Adding to this, what I often see lacking in essays is the conceptual jump from the specific activities and experiences you have had, to what you learned from it or impact it had, and how that can be applied to your practice of medicine.