Good question! To "master" something usually means you know everything about it, you can do it very well, and implies you have worked hard on it for a long time, but when you "get the hang of" something, it usually means you are starting to get good at the basics of something, like you are still a beginner but are starting to understand what you are doing.

"to get the hang of <doing something>" is a slang expression. It means "learn how to (do something)". We often use it when we talk about learning how to do something successfully or to an acceptable standard. "To master (something)" is not slang. It means "to learn (something) to a very high standard". Mastering something describes a much greater degree of skill than getting the hang of doing something.

(1) means that after years of practice, one has become an expert at weaving. (2) means that after years of trying, your are finally able to weave acceptably well. "to get the hang of" means to learn how to do something. You can say, "He finally got the hang of riding a bicycle without training wheels, and now he can ride without falling over." Or, "He finally got the hang of typing without looking at the keys." (1), on the other hand, means that you are very good at doing something. You can be called a master or expert.

“Master the art of” implies that you have become very good at something. “Getting the hang of” implies that you have recently learned it, but can still get much better at it.