Someone who's snobbish makes it clear that he thinks he's better than most other people. Your snobbish cousin might refuse to borrow your rain coat because it looks "cheap."
If you're snobbish, you believe there are different social classes of people, and that you belong to the highest one. A snobbish person might think that graduating from a certain college makes him better than others, or that having more money makes him superior to those with less. The root is snob, which evolved from meaning "shoemaker" in 1700's Britain to being university slang for "ordinary person who apes his social superiors" to "one who despises those he considers inferior."