Carefulness is a characteristic of being cautious, thoughtful, or prudent. When you write "FRAGILE" on a package you're mailing, you're hoping it will be handled with carefulness at the post office.
In many cases, you can substitute this noun with the shorter care. But in some cases, carefulness is more accurate, implying "caution" rather than "loving protection." While your best friend is a daredevil, your carefulness in all situations makes the trampoline park slightly less fun for you. The Old English precursor to carefulness, carfulnys, originally meant "anxiety."