There was another post about who gets more respect Marines or Army infantry and in my answer I said that as a former Marine I respect all members of the armed forces, I chose the branch that I felt was best for me,and that's what you should do.
Whichever you join you just make sure you are the best soldier you can be and that will be all you need to know to fill you with pride. I wont put up with anyone disrespecting any branch of service. Think about it. Can you remember the beaches of Normandy and have anything but respect for the men who fought and died there? Sure I am proud of my Corps, but I have known some damned fine soldiers that could cover my back anytime.
I guess what I'm saying is, forget about how much respect you think you will get from others. Earn self respect by training hard and always doing your best. Then you wont worry about what others think in ignorance because YOU will know.
I don't quite understand why you say either Army infantry or MarineMP's. Why not Army MP's or Marine infantry? ( Actually I never cared all that much for Marine MP's. They were about as chick$%#* as they come) To some extent what you say about the Army getting better equipment has historically had some truth to it. But Marines have their neat toys too.
I would suggest you go to the library and check out some books and read a little bit about both branches of service. Maybe some history about both branches in combat. See if you get a better feel about what each has done and get more partial to one or the other.
I promise you that whichever branch you choose you wont regret your choice once you're in. Marine Corps boot camp is a challenge and completing it is going to bring you out of your shell a bit and give you a lot of pride and confidence. If you think Army basic isn't enough of a challenge enlist for airborne then try out for Ranger school.
Last of all remember that the most important characteristics of being a good fighting man are intangibles that you can't learn in a few months. Courage, honor, a sense of duty, responsibility, all come from character. Wisdom and maturity are important too. So make a wise and mature choice.
Good luck.
Semper Fi