It sounds like a "career diploma in cosmetology" is certification that you have completed cosmetology school. The number of hours it takes to complete cosmetology school will depend on your program of choice, the state you're in, and whether you attend cosmetology school full time or part time. However, once you have completed cosmetology school, you still need to take and pass the state cosmetology license exam in order to become a licensed cosmetologist.
With a Cosmetology license, you can work with hair, skin and nails. You can do hair, you can also work in a dermatology office. The options are endless. Along with a cosmetology license, with further training you can be a master hair colorist. Like I said, the options are endless. Like every other license, it gives you the right to perform any job related to your specific cosmetology license. Hair styling for cosmetology hair stylists, and etc.