Vegetarians don't eat animal flesh of any kind. No beef, no pork, no chicken, no fish, no other seafood. No rabbits. No birds. No wild game. But they may eat dairy and eggs, and they may use animal products like wool and silk. Whether vegetarians wear leather is controversial. Some people say that vegetarianism is solely about what you EAT, therefore you can wear leather and still be a vegetarian. Others say that leather is as vegetarian as a steak-- either way, you have to kill a cow to get it.
Vegans take their moral commitment a step further and don't use any animal products of any kind. All of the above, PLUS, no dairy, no eggs, no wool, no silk, definitely no leather, etc. It's a lifestyle choice as well as a dietary choice. There appears to be some controversy about the vegan-ness of honey, but I don't have a dog in that fight, so I have no comment, except to point out that insects are animals.
You have to understand, too, that "vegetarian" and "vegan" mean something very different to many people outside the vegetarian/vegan community, because they don't understand it either. A lot of people think vegetarians eat fish, and some of them even think vegetarians eat chicken. And many, many people think "vegan" just means a vegetarian who doesn't eat dairy or eggs. True, that is the most high-profile aspect of being vegan, but there is much more to it than that.
One of the terms I've heard used to describe a vegetarian who doesn't eat dairy or eggs, but is not completely vegan, is "strict vegetarian."