Profane means non-sacred or outside the temple—thus, irreverent or blasphemous. The term is also used for general verbal expressions that sound or are intended to sound insulting, rude, or disrespectful.
To blaspheme is to speak irreverently about sacred or holy things (apologies for the circularity). Blasphemy has a religious component. Some countries ban blaspheming.
Obscene means outside the standards of decency and morality, and often is applied to sexual matters. What is obscene is contextual. And good luck with a universal distinction between obscenity and indecency.
Curses, strictly speaking (haha), invoke supernatural power to inflict harm or punishment on someone or something. Cuss is an informal version of the word, in this context.
Swearing is to make a solemn statement; it’s also to use offensive language, often in anger.
Vulgarity is an indecorous expression specifically chosen for its unrefined character; it generally has a social and moral component, even if casually chosen.
Okay. Now, don’t confuse or blur these!
And note that they don’t all overlap. Remember this when listening to the current hubub about the Supremes’ latest decision on language and broadcasting. And remember that profanity, obscenity, etc. are contextual, both in the moment (who are the speaker and audience? what is conventional for them?) and within the social experiences of the user of the language.
So, “swear words” may be profane, obscene, vulgar, or curses, but rarely are all of them. Most common swear words in the US of A these days are obscene and vulgar. A few are profane and curses; to some, that makes them also vulgar. But, the overwhelming majority users of these words do not think of any of the common swear words as blasphemy—and may not be able to come up with any words they think of as blasphemy. See how complicated it is?