I have dewalt tools and Milwaukee tools, i also have an adapter so i can use my dewalt batteries for my Milwaukee tools, I can not say which tool is better, both are great. but if i had to pick one, i'd go for Milwaukee,
As far as adapters, they are all third party aftermarket product, no tool manufacturer makes these cross brand adapters, dewalt makes one to be used in dewalt 18v nicd tools to be used with dewalt 20v li ion battery. The rest are made by unknown companies, and vary in quality, some may have a circuit inside, others not so much. pretty much all adapters only have 2 pins, so you can not charge a battery with that adapter on it. Some have wires so thin that they smoke insulation with a power hungry tool. some are made on 3d printer and shatter if accidentally dropped together with the tool and a battery. those adapters with circuits, sometimes would drain your battery even when not in use, so remove it from the tool when you are done working with it.