Vegetarian is simply the non-eating of meat. What is considered "meat" varies from diet to diet as some people do not considered fish "meat" and some do not consider fowl "meat". It is a dietary choice that is very old and can be healthy if done right.
Vegan is no animal products at all. So no honey, no dairy, no unfertilized eggs which are all permitted on a vegetarian diet. Their view point is that eating any of those things is "exploiting the animal". BUT they will happily chow down on food that were pollinated by insects and never worry that this is exploitation. Vegan is a new religious movement adopted by weirdos.
Vegetarians can be kind of a pain if they are the evangelizing type. You generally find that type in the new converts. They often go on to become vegans because they are trying to convince themselves that they are "nice" people.
They aren't.
I remember seeing news reports of a demented couple, who killed their baby by feeding it nothing but a vegan diet.