"What's the difference between Vegan and Vegetarian"

-Vegetarianism is a dietary choice. A vegetarian diet does not include anything that doesn't come from the plantæ/fungæ kingdoms, or Cynaobacteria. Vegetarian: Vegetable + -arian. Let's now define vegetable: a plant or part of a plant used as food, such as cabbage, potato, carrot, or bean.

"So, do eggs and milk fit in there?"

-Well, milk comes from mammals, from their mammary glands. Where some plants produce seeds (fertalised eggs), which are similar to eggs found in the Animalia kingdom, they're not. Eggs have a shell and an amnion. With shell and amnion, they are found in reptiles, birds, and monotreme mammals. Eggs are also found in fish and other mammals, such as ourselves and marupials. In the case of us placental mammals, they never leave the body. So, no, Vegetarian diet does not include eggs or milk, because they are not vegetables.

"Why do people assume those two things are included then?"

-Most likely it's because they don't pay attention to definitions. There is a reason that lacto-, ova- prefixes exist. They are additions to words, such as vegetarian. So, a lacto-ova-vegetarian consumes these other reproductive products and byproducts in addition to the vegetarian diet.

"What's a vegan then?"

-Well, a vegan is someone who adopts a vegetarian diet, without additions to it, and also avoids or excludes all animal byproducts and products. E.g. A vegan does not wear leather or wool or consume honey, or buy products that have been tested on other animals. It is a lifestyle choice.

"Is Fish a vegetable?"

-No.