A vegan diet only means a diet that doesn't contain any animal product and no there is no nutrient you could get from animals that you can't get without them. If you disagree, give me one and I'll tell you how you can get it.
Well, there are quite a few nutrients that a vegan will not get on an organic diet, or that he won't be getting enough of. First of all vitamin B12, this can nowadays be found in vegan cheese, but that won't be organic due to supplementation.
Vitamin D is also almost non existant in organic vegan food, but nowadays you can find it in fortified stuffs. There are two vitamins used for fortification: D2(from yeast) and D3(from animals), however our bodies only have about 60% the efficiency of vitamin D3 when consuming D2, so you'll need to increase your intake.
There are also carnosine (an antioxidant), which can be mostly replaced, but not completely, by eating other antioxidants, and DHA can only be obtained by eating fish. Our bodies, as advanced as they are can still convert some other omega 3 fatty acids to DHA although it's not as efficient, and the efficiency varies from person to person
There are major healthbenefits gained if compared to those that don't balance their food intake correctly, however the healthiest people in general eat both in a balanced manner.
I'm not really against anyone keeping himself from the tasty stuff (yay more for me), but they need to be careful about balancing their nutrition correctly. Also if done over a longer period of time, the body sometimes looses some of the digestive functions (only temporarily) so they will have some stomachaches and stuff when they try to eat meat again.