I have some experience to make a comparison here because I was in Army JROTC in HS and I ended up joining the Navy. During my time in the Navy, I worked in special warfare and had the chance to work with Army personnel many times.
It is the general consensus that Navy personnel (this includes the Marine Corps), on the whole, are treated better than Army personnel. I take this not only from what I saw in person, but also from the testimony of soldiers at Fort Ord, and those that I worked with on joint ops. Navy people generally eat better food, sleep in better quarters, and seem to be treated with more dignity. That being said, regardless of which branch you join, there are always going to be people who are anxious to throw their weight around and remind you that they have a higher rank than you, just like your dad did to that Army guy at the exchange. That, by the way, is a big reason that a lot of people get out after their first tour. They get sick of being pushed around by people just because of a difference in rank.
No matter what you do, here are some things that I can guarantee you:
If you join the Navy, you will be able to negotiate for your orders. Of course the needs of the Navy always come first, but you do get to negotiate with a detailer in Washington who will give you some choices. My understanding is that, in the Army, you turn in a wish list and hope that they pay attention to it.
If you join the Navy you will get a broader base of training than in the Army. This is obvious, based on the fact that the Army has so many more job titles. I worked in communications when I was in the Navy, and I would operate HF, UHF, VHF, and Satellite communications when I was in the field. The Army equivalent to what I was doing was four separate people (one for each frequency band), each with a different job title. Of course, in the long run this also meant a lighter workload for those Army personnel.
If you join the Navy (or the Marine Corps) you can bet that you will end up spending some time floating around on a ship with nothing to do. This is not always the case, but usually happens. This will probably happen in the Army as well, but at least you would be based on land with some chance at getting liberty during that down time in the Army.
My advice is to talk to recruiters from all four branches and see what jobs they are offering you, based on your ASVAB score. Think about the future you might want to have after the military, and see how well that training might suit that future. Also consider college options and, if you are thinking about college, see if you can get some sort of college fund above and beyond the GI Bill.
Hope this helps.