I would say that the only difference that you have to overcome is the tight-knit community of alums. As a CS/OR major from a Top 4 school, I haven't ever gotten an internship offer simply because I was at the school I'm at. Stanford especially is good at fostering connections in the startup community for their students, but Princeton has NY and Harvard has Boston.

What is an OR major please? Google didn't help.

I don't know about Stanford, but in several other schools, it stands for Operations Research.

Yep, ayla has it right, Operations Research. A bit similar to applied math.