I think they are basically close to being the same. I, myself, like a diet that includes ALL food groups, this way you get all the needed vitamins, minerals and amino acids. However, if I were to do one, it would be vegetarian, there is more of a variety of food where the vegan is ONLY vegetables with nothing from animals such as dairy product.
If you do either one of these, make sure you get the proper amount of B vitamins, especially Vitamin B-12, this is not found in the true vegan diet at all and very little in the vegetarian diet.
Good luck, let us know how you are doing