Does one US military branch think it's superior?

Each branch has that feeling simply because of the rivalries between services. But the truth is that we all know we are part of a team and each branch has a specific job to do and mission to accomplish to make everything work smoothly. We are all brothers in arms and that's what makes us unique. I am proud to call everyone here in RP my brothers and sisters as well as all who are and have served.

Sir, I think I'm the wrong one for this.....objectively yes it serves a greater good to have the forces working in a "joint fashion". It saves money, time, and confusion. It also allows commanders means to double the forces power, bring in alternative viewpoints, and it gives someone who like writing regs a job. I see it like this:
1. So you personally want to see it destroyed? No? Join the Navy
2. So you personally want to destroy it? Yes. Join the Marines
3. So you personally want to destroy it and hold the continent as a prize? Yes Join the Army
4. So you personally want to see it destroyed from the air? Yes Join the Air Force
5. So you REALLY want to see something on a massive scale be destroyed, obliterated, and scraped off the face of the Earth so no one will ever remember it? Yes but not be there? Join the Army but go no higher than SPC, or become a Warrant Officer. Both can accomplish the task, with speed, precision, and skill, and no one will ever know they were there.
For those who doubt #5, ask this, the Green Berets have Warrants on the teams. How often do you see him? More importantly how often does he actually see you? Warrants are everywhere but nowhere at the same time.
This is in humor folks, I give a laugh and I hope to get one in return. My meme is for my Submariners. Much respect to you, even though the E4 Mafia can do the same thing.

I never once felt superior because in the Air Force we could all read & write.

They're all superior in their specific area of warfare. So yeah, we are superior.
Army: superior at holding ground
Marines: superior at taking ground
Navy: superior at naval warfare and bombing the shit out of the enemy at close range
Air Force: superior air power and bombing the shit out of people at long range
Coast Guard: superior finding drugs, refugees, and saving live and all the other stuff they do

COL Mikel J. Burroughs no rivalry at all, actually, I would love to go to their obstacle course...

Not only is there an "inner service rivalry" but within each military service their are rivalries COL Mikel J. Burroughs. Within each service the ground and air components have rivalries for resources, training and operational employment.
Part of this is good-natured competition; but, the budget process is competitive and tends to discourage efficiency within an organization.
Some people will always consider themselves "better" than others. Since we are each created in the image of God we are all valuable as individuals. I have long done my best to teat each person with respect independent of their rank or position. My respect was not always returned; but, I know it is better to follow the command of Christ to to do unto others what i would like to have done to me.

All good humor If you heard secure the building what does that mean," The Joint Chiefs once told the Navy to "secure a building," to which they responded by turning off the lights and locking the doors. The Joint Chiefs then instructed Army personnel to "secure the building," and they occupied the building so no one could enter. Upon receiving the exact same order, the Marines assaulted the building, captured it, and set up defenses with suppressive fire and amphibious assault vehicles, established reconnaissance and communications channels, and prepared for close hand-to-hand combat if the situation arose. But the Air Force, on the other hand, acted most swiftly on the command, and took out a three-year lease with an option to buy." Mind you I am Air Force.

I throw humor in it to say this. The Air Force is good at Air, Space, and Cyberspace. The Army is good at taking and holding ground, the Navy is good a projected firepower a couple of miles of the shore very quickly and the Marines are good at taking ground in any environment. Of course we all have special forces that are unique as well...not superior, unique in their missions. There are none that are inferior or superior...just better at what they are specifically called to accomplish. In times of dwindling budgets, there are also growing opportunities for joint ops that create mission success exponentially. I have witnessed this first hand. Navy corpsman triaged me in the deployed outpost clinic, Army and AF docs diagnosed, Marine and AF led the convoy to get me to the hospital and a Army medic stayed by my side the whole ride out. So I guess you could say that I believe in joint ops and that they all are good at what they do without superiority.

There will always be an inter service rivalry just like a brother to brother rivalry. BUt no body else better pick on my brother because when the proverbial poop hits the oscillating device the service names go away and we fight as one.

Oh yes! 
Even among the MOSs:

And the Lord spake forth unto the heavens and said, 
"Let there be Airborne." 
The earth did tremble and quake 
and the waters did rise up 
and the clouds did part 
and there came forth a multitude of parachutes that filled the sky. 
God looked down and saw this was good. 
And they were good; they were AIRBORNE. 

God then spake forth unto the land and said:
"Let there be Rangers." 
And all at once the day turned into darkness, 
and the winds did howl, 
and mountains crumbled into the sea, 
and the great rocks did part, 
and there sprang forth a horde of Mephistopheles' disciples 
wearing Ranger tabs and carrying all sorts of deadly weapons. 
God looked down and saw that this was bad. 
And they were bad; they were RANGERS. 

Then God spoke forth thrice, unto the sky, the earth, and the sea, and said: 
"Let there be Special Forces." 
Lightning did flash and echo across the sky. 
Mountains spewed molten rock and rained fire upon the land. 
Tidal waves surged against the shore. 
Dispair, disorder and turmoil did prevail. 
Forthwith, there did appear a band of twelve extraordinary men. 
A few came from beneath the waves; 
others jumped from the sky; 
and more still stalked silently from the dense forests. 
Each one was in camouflage battle dress, wearing a Green Beret. 
Working together, they brought peace unto the land. 
God looked down and saw that this was amazing. 
And they were amazing; they were SPECIAL FORCES. 

Besides himself, God now spake forth again and commanded: 
"Let all ye that be weak in mind and body, arise and go forth." 
And lo, from the abyss they crawled forward with indecision 
and limped weakly upon the earth. 
God looked down and shook His head, for this was pathetic. 
And they were pathetic; they were LEGS.

Sir, in the states, there was good natured rivalry. In Vietnam, all the branches worked together to accomplish the mission. Now as a veteran, I look at other veterans as just that: fellow veterans, regardless of branch of service.

Seriously though, we all have our respective roles to fill in the Big Picture. It all boils down to each service component has a mission set it does better than anyone else. The secret is knowing which asset to utilize for the best result or intended goal. I was privileged to work in several Joint units. The Army, Navy, Marine, and even the Coast Guard troops were some of the best I've seen at their respective tasks. I consider it an honor to have served with each of them!