The word "twat" was used, especially in the U.K. centuries before it became a slang term for a woman's genitals. The OED cites the early 1650s as the date of first use, and the word was a derogatory expression used to describe someone who is dumb, stupid, or careless. It wasn't until the early 1900s that the first began to used with it's second meaning, and it wasn't until the 1950s that its use became widespread. The British still use the word "twat" in derogatorily labeling someone an idiot.
Meanwhile, the word "cunt" has been used since the early 14th century, and it's been around, in its present meaning, for close to 700 years. In fact, Shakespeare sneaked the word into "Hamlet," albeit in a somewhat disguised fashion. Ironically, while "twat devolved" from a general to a specific term, "cunt" followed the opposite course, moving from the specific to the general, e.g., "She's such a cunt."
Which word is worse? In general discourse, I think "cunt" is worse. Despite it's etymology, "twat" is a relatively common slang/vulgar term. It's a course word for vagina. "Cunt," on the other hand, holds deeper meaning, referring both to the body part and the person with that body part.
So, where does "pussy" fit into all of this? While it's a word I do not use myself (except in a very strict context), I think that "pussy," while still vulgar, is the least offensive of the three words. Perhaps akin to the difference between "penis" and "cock" or "dick."
Okay, I understand you didn't want an etymology lesson. But MissBarbara is gonna MissBarbara...