I grew up around East Coast schools and I can say, yes, there is a difference.
I attended Texas A&M as a graduate student. I lived there, I went to classes, and I taught labs of undergrads, so I have somewhat of an insider (but outsider) viewpoint.
I see Texas A&M as a more concentrated subset of Texas culture. From what I can tell, kids growing up in Texas are taught (I assume since birth) that Texas is the best state ever and that Texans are the best people ever. Anywhere else is just downhill (their case is somewhat validated by being surrounded by Arkansas, Louisiana, Oklahoma…)
I also presume this is what occurs at A&M. From the beginning, all freshmen go to “Fish Camp” which is a several day camp where they learn all the traditions, cheers, etc. My wife and I jokingly call this “indoctrination camp” because this is where we presume that they all get told (repeatedly?) that Texas A&M is the best university ever and all the other universities are lame. All the other students in all the other universities know this but won’t admit it. As a reiteration, when you go to college and have all this school spirit and great experiences, that only happens at A&M!
Of course, I exaggerate somewhat, but the effect is there. This is subconsciously reiterated throughout your time at the university and so it somewhat sticks.
I think the history and the strong emphasis of the Corps of Cadets plays a role. Also, the university has a lot of group traditions that bring the student body together and helps to create a defined cohesion.
Taking all of this into account, A&M is an average, large, top-30 public institution of learning, with nicer than average people and uglier than average campus and buildings.
Before all the Aggies jump on me and rip everything I said to shreds, I really enjoyed my time there. The people that we me there were wonderful, and if I had to live in a medium sized city, college station would be a great choice.
Edit Add: One of my current coworkers grew up in College Station (her father was a prof) and she mentioned that she thought it was very cult-like.