The difference between a vegetarian and a vegan is not so simple as you may have thought. Basically there's a difference in food that vegetarians and vegans eat, but it seems that sometimes there is a deeper pattern involved - beliefs. To make these considerations clear we have to start with few definitions.
Vegetarian - term used to describe a person who doesn't eat animal flesh. We can divide vegetarians into few smaller groups. These groups are:
Vegans - the strictest group of vegetarians. They do not consume any animal-related products. There are some problematic questions between vegans. Two of them are: whether or not to eat honey (it's made by bees) and bread (it's made using yeast). A good answer to those questions probably doesn't exist so it rather depends on a person whether one feels like consuming those products.
Lacto vegetarians - a little less strict group than the vegans. They allow consuming dairies like milk, cheese, etc, but abstain themselves from eating eggs.
Ovo vegetarians - also a little less strict group than the vegans. They allow consuming eggs but abstain themselves from eating dairies.
Ovo-lacto vegetarians - they allow consuming both eggs and dairies.
As you can see every vegan is a vegetarian but the opposite situation isn't always true. People become vegetarian out of a few reasons. One of the reasons is their health. Some vegetarians are driven purely by the sake of their well being. Some people just take the vegetarian diet for a test drive and after some time they conclude that it's beneficial for them, so they stay this way. Because of the health motivation those vegetarians don't seem to be as strict as vegans. They allow eating eggs or dairies because it's profitable for their health. The other reason leading people to become vegetarian is ..
In some cases moral choices are the most important factor for people choosing to become vegetarians. Their consciousness imposes them to abstain from animal-related food. When personal beliefs comes into consideration people tend to be uncompromising. Because of that they mostly choose very strict forms like veganism. They don't treat it only as a diet, but something much more important in their lives. Some people even say that being vegan isn't just a diet, it's a lifestyle. Besides food vegans often refrain from using any animal products like the cosmetics that contain animal ingredients or were tested on animals. Some of them go even further - they volunteer in organizations fighting for animal rights.
As you can see the difference between a vegetarian and a vegan isn't only based on the food which certain sub-category of vegetarians consumes. It's OK if someone is an ovo-lacto vegetarian because of one's moral choices but one can as well become ovo-lacto vegetarian because of health reasons. On the other hand being vegan is mostly caused by someone's individual beliefs and often draws along many other changes in lifestyle besides eating habits.