I had a hard time putting this into a single term because I am not talking about championships or wins, but the larger spirit of athletics. Cal students know that nothing is handed to them and you fight for everything. What does not kill you only makes you stronger... in that same vein, Cal athletics represents that spirit.
Regardless of the sports, Cal teams never quit. They always represent their school proudly and show up ready to play. I think this attitude is reflected in the general student body.
Similarly, Stanford athletics is reflective of their students. They feel entitled to things in life; coasting through undergrad with little regard for hard work. When the going does get tough (I assume that mean getting a "B" in a class) oftentimes they fold.
Here are two recent examples of the Stanford attitude:
1) The Stanford football team has been plagued by reports of players quitting. Leading tackler Michael Okwo was reported to have quit the team but then rejoined. Even when Cal was 1-10, the team played hard and never embarassed the school.
2) Stanford Rugby team forfeits match against Cal Rugby. This best exemplified the attitude of Stanford and its athletics. Greatly overmatch, the Stanford team choose to take the easy way out rather than man up and play. With the amount of monumental upsets that occur in sports every year, it take quite a group of quitters to think that they have no shot of winning. I imagine when the Stanford rugby players asked their friends on campus if they should forfeit the match, the overwhelming response was, "Sure, no shame in quitting, I just dropped 4 classes after I took the final."