Current senior here who attended both Girl's State and NJROTC Leadership Academy (LA). I am going to be adding more of my insight on LA for two main reasons 1) I am hoping to give you a well-rounded interpretation of both camps by making up for the lack of knowledge on LA simply because it is the lesser-known camp and 2) I was selected to return and as a cadre (platoon advisor/instructor assistant) meaning I have a behind the scenes look at the overall goals the camp strives to instill in its cadets.
I apologize in advance, this is pretty long but from what I have seen, this is a decision many have to make so I hope it helps you and anyone else dealing with the same problem.
First off, huge congrats to you, and anyone in the same boat, for being considered to attend these camps. Many strong candidates to the SAs are where you are, meaning you are definitely on track. Both are incredible opportunities that I full-heartedly believe you will not regret your decision one way or another. That being said, I do feel as though each camp will attribute greatly to different parts of your resume.
Boys State: While I attended Girl's State and the two are quite different in a few ways, the general goals of both camps are the just tailored slightly to the attendees (boys vs. girls). The end result of the camp is, according to my experiences, to instill patriotism, understanding of the US government and to build camaraderie and create a state-wide network of top-notch boys/girls. These ideals are what the academies are trying to drive into their candidate pool. The main pros coming from Boys State, from my perspective, are becoming part of the network in your state, getting a hands-on experience on the inner workings of the government and most importantly demonstrating your interest in the academies as they all recommend you go. (I would add on but there is already so much that has been said on this thread that would just end up being repetitive)
Leadership Academy: While I'm not sure which Area you are in, the camps are designed to be extremely similar all across the country. LAs are there to give the top cadets an opportunity to be put in a high stress environment and learn leadership through both classroom and hands on learning through interaction with others and their own mistakes, all to develop them further as leaders so they can return to their units in the fall and improve them. While these camps aren't specifically listed by the academies, they have so many applicants from NJROTC, that they have definitely heard of it before. The main pros coming from LA are the leadership training and experience and the networking throughout the program. I cannot stress enough how much LA changed me. I learned so much about interacting with others, dealing with all different sorts of situations, and picking myself up after the failures that no doubt will plague your week by design. These examples and experiences gave me so much to draw from during the numerous interviews of the application process. I would highly recommend LA for that reason alone. These also give a tremendous networking opportunity within the program and while being known by other cadets is great, but what mostly counts is by those who can write you a recommendation letter. My own Area Manager wrote me a very lengthy recommendation letter to all the academies I applied for because he knew me by name. The academies want to see you commit to and extra-curricular and be successful. LA, I would say, is the route to take if you depend on having NJROTC as a strong part of your resume.
Decision: I believe both are great opportunities and I think the answer to the question: Leadership Academy or Boys/Girls State, is subjective to each individual candidate. My advice would be to pick the summer camp that will make you the most well rounded. I feel like Boys State will contribute more to your resume on paper as its encouraged by the academies while LA would contribute to your skills/experiences and not have as many points on your resume but would definitely come across during your interviews and even your confidence. I believe most would benefit more from Boy's/Girl's State however if LA is what you want to do, don't let that keep you from attending. Again, this is all my opinion based on my experiences in both camps.
In the end the decision is yours, and there is no wrong decision the way I see it. Consider which you are more excited to go as this is likely a great indicator of which camp you will put more effort toward. Either way, I cannot stress enough how important it is that you take full advantage of whichever camp you decide to attend. If you end up going to Boys State, write a bill, apply for positions, and make good friends you meet, they are the kind of people you want to surround yourself with. If you go to Leadership Academy, challenge yourself to step outside of your comfort zone and don't be afraid to make mistakes, just learn from them and hold your head high: they are proof you are giving 100%.
Best of luck to whoever reads this and know that ultimately this is a great problem to have: "which selective summer camp do I go to?"