A vegetarian diet means you do not eat meat. A vegan diet means that you do not eat animal products of any kind, including eggs, fish, milk, dairy etc. I have some vegan friends and they won't let their daughter eat skittles because they contain gelatin that is made from some kind of animal product. Their daughter is 12 and very small, but smart and healthy. They say they have been told by their doctor that vegan children can be exceptionally small and will grow and mature late. But, they will eventually reach a normal size. For what its worth they are not liberal at all but are eating this way for health reasons.
That's the same with my (Conservative/Republican) brother and his wife.
My niece is being raised vegan and is very healthy. But she has two educated parents who know what to do and eat a huge variety of foods etc.
Strict vegans won't eay honey or wear a wool sweater, either, so as not to enslave the bees and sheep.
There are fruitarians, also, who won't kill a vegetable but will only eat fruits and nuts after they've matured and reproduced. I once saw a bumper sticker that said "Mowing is Murder". Apparently you have to let the grass spread it's seed before it's moral to cut it. I'd be laughing but I can't stop thinking of those poor kids.
There are essential fatty acids and other nutrients that just aren't available from non-animal products, and others which are only available as synthetic supplements which 1) many extreme "natural" diet people won't use either, and 2) sort of neutralize any claims that this is natural way for people to eat. The specifics vary somewhat by ethnic background -- certain people of scandinavian and Inuit descent simply cannot synthesize the essential fatty acids found in fish, while other people can (fairly recently, a DHA supplement made from algae, which is where the fish get it from, has been introduced, but it comes in a gelatin capsule, so extremists won't take it). Other groups simply need more or are able to synthesize less of various essentials (ethnic Africans living in non-tropical climates have serious vitamin D issues, for example). Vitamin B12 is an issue for all people, though synthetic supplements can do the trick.