The main difference between both is what I am going to tell you here. All vegans are vegetarians and much more than that, but not every vegetarian is a vegan. The difference stems from the fact that both vegetarianism and veganism definitions stipulate what products can be used and what cannot be. This means eating or using anything other than the product that comes under the definitions can be the difference between one being a vegan and a vegetarian.
 
Nevertheless, when it comes to veganism, it is worth noting that using something non-vegan by mistake is not the end of the world. This is because its definition takes the maximum viability of the user into account. For instance, how could you tell for certain whether a drug has been tested on an animal beforehand unless the manufacturer reveals it? Unlike veganism, the definition of vegetarianism does not cover these minute details.