In a casual conversation, both sentences mean essentially the same thing: the second person is not upset or inconvenienced by not being able to wear their hair up. They are fine with not wearing their hair up. Most people will understand the meaning of the two sentences as being similar/the same. There IS a subtle difference: I am not that bothered about wearing it up. = this sentence implies the speaker has a weak/slight preference for wearing their hair up, but is not troubled or upset about the situation. --The key element is the expression "not that," which is a way of modifying the intensity of the meaning of what follows. I do not care about wearing it up. = this sentence literally means that the person does not have a preference to wear their hair up. Strictly speaking, the meaning of the sentence is vague about whether the second person prefers to wear their hair down, or simply has no preference about wearing their hair up or down. Please note that this difference in meaning is VERY subtle. Unless you are studying English for a goal of advanced/university level fluency, mastering such a subtle nuance in meaning probably isn't necessary. So don't feel bad if it is confusing. Some other examples using the "not that" modification: "Well, she isn't that pretty." = the speaker is willing to admit the woman is pretty, but is trying to minimize the extent of her beauty, or to be a little bit mean. "The movie wasn't that great" = the movie wasn't bad, but wasn't particularly good either. In fact, it usually would mean that you didn't really like the movie. Good Luck!

My understanding is that bothered means you don't take the action of putting up your hair because it doesn't have priority or your focus. Like "I couldn't be bothered from studying to play games." Studying has focus, but you still care about playing games. To care, could be, you won't let the action effect you. You can not care about putting up your hair, and when people judge you, you are uneffected by their opinion. They are synonyms, so they are very similar. Let me know if I can clarify anything.