Pros: Microwaving is the quick and easy option when cooking vegetables. Since less water is used, you're left with more nutrients in your food than when boiling. It's also said that you're left with more flavour and colour than when you boil vegetables.
Cons: Microwaving is seen as an unnatural way of cooking. It's not recommended to microwave vegetables from frozen, as the longer it takes to cook something, the more nutrients it will lose. Also, frozen foods tend to cook unevenly. Make sure the containers you put your food in are microwave friendly - It's believed that if the container isn't, it could contaminate your food.
Pros: The traditional way to cook your vegetables. It's recommended that you keep the water you boil your vegetables in - this water contains lots of nutrients which can be used to make gravy or a nutrient-filled soup.
Cons: When boiling, the nutrients in the vegetables will be lost in the water that you drain.
It you're microwaving fresh vegetables, you'll keep more nutrients by microwaving them but boiling is advised for frozen vegetables as the boiled water can be beneficial if making gravy or soups.
Currently, the best way to keep all the nutrients in your vegetables is by steaming as it doesn't alter your food as much. Steaming can be done in the microwave or on your stovetop.