assuming both diets follow proper nutrition, including vitamin supplements if necessary

Not necessarily, I have a vegetarian friend and he weighs more than me. You lose weight by eating less calorie than you give out. A lot of vegetarian food use flour as a substitute of meat. It's high in calorie. If you don't eat enough meat, you might suffer from protein deficiency. Doesn't matter if you're a vegan or not, as long as you avoid sugary, greasy and junk food, as well as alcohols and soft drinks, your diet should be healthy enough.