The two don't have anything to do with each other. Vegan, as see said, means no animal products or by-products (and you have to look pretty deep, some animal products aren't as easy to spot as others) while organic (as used in marketing) means that something is grown/raised/produced with no synthetic pesticides, fertilizers, etc.
There's more info on the definition of organic as used on labels here:
I am vegan and although I grow a lot of my own food using organic principles, I can't always buy organic produce due to availability.
Being vegan is about living without using animals or animal by-products. That includes not using or wearing products that have leather or silk in them. Not eating or using products that have bees wax in them or any egg products.
When using skin care products etc ... it's about using products that are made from vegetables, fruits and nuts.