Research shows that vegetarians and vegans have better heart health, and have a much lower incidence of diabetes than their meat-eating counterparts (provided they eat healthily). These are not to be sneezed at, as they are two of the leading health problems in the Western world right now.
Vegetarian/vegan risk profiles
There are multiple research studies and papers pointing to vegetarians and vegans being at higher risk of things like stroke, poor brain health, hair loss, stress, and possibly mood disorders.
The majority of these issues arise from misunderstanding the proper nutrition your body needs, and not adequately replacing the vitamins and minerals that meat would have provided. Most of these issues can easily be corrected with a well-balanced vegetarian diet and if necessary, some dietary supplements.