Vegan = no animal products at all. Vegetarians avoid eating animals themselves but can be OK with animal by-products like dairy and eggs depending on how strict a vegetarian they are.
In terms of vegan baking, the main things you have to watch out for are dairy, eggs, gelatin, honey, and cane sugar (which is processed with animal bones, beet sugar is OK). There are many different egg replacers available (flaxseed, banana, baking soda/vinegar, potato starch, etc.), and you can sub soy-based margarine and soy milk for butter and regular milk.