The quick answer is… it depends.
A junk food vegan diet, which is rather common among new vegans who are active with animal rights issues, is not healthy. And while you will earn the karma of grateful cows, it won’t do much for your physical health.
A whole-foods, plant-based vegan diet, that is high in vegetables (especially green leafy ones), fruits, legumes and whole grains will lead to improved health, lower blood pressure, lower cholesterol and lower risk of heart disease and cancer. An added bonus, it is helpful for weight loss.
I highly recommend watching the movie Forks over Knives for some powerful examples of lives that were changed by switching to a whole-foods, plant-based diet.