Kindle Fire HD 7" vs. Nexus 7 — GO!
Okay. It's fair to say that this isn't really a question of better, or of a winner and a loser. Each tablet is going to have strengths and weaknesses that make it a better choice for the needs some users over others.
The real question is, what are those needs, and how do the strengths and weaknesses apply? These two devices are arguably very close in specs and price, and so there's quite a bit of overlap in the potential user base.
In short: You have $200-$250 to spend. You know you want a 7" tablet. Which of these do you pick, and why?