Are there any drawbacks to giving my one-month-old baby water in-between feedings?
Yes, there is a drawback. A one-month-old spends a lot of time and energy eating, and in return, should get the calories, vitamins, minerals, and protein needed to grow. Water provides fluid only, so an infant spends that energy without getting calories to turn into more energy. Even on the hottest of days, when you are reaching for extra water, extra breast milk or formula provides enough 'free water' (they are more dilute than most people realize) to compensate for extra fluid needs.
Is an occasional few ounces of water on top of a healthy milk intake going to cause harm? No, but neither is it a necessary part of a one-month-old's diet.