But the difference might not be between Harvard and a local community college. People don't come out of "slums" to Harvard and immediately join the upper class.
But you'll be exposed to different people and opportunities by going to a state school instead of a community college, or a better school out of state with a scholarship vs. a state college, etc. The small differences matter.
I literally got my first job because my roommate's dad knew of someone looking for a software developer job. Would I have met that person in community college? Maybe! Lots of people with good networks go to community colleges, too - but I'd bet fewer than more expensive schools.