"Flash Fiction" is an umbrella term used to describe any fictional work of extreme brevity, including the Six-Word Story, 140-character stories (also known as 'twitterature'), the dribble (50 words), the drabble (100 words), and sudden fiction (750 words). Is the genre significantly different from poetry and short stories, and, if so, what are its distinctive attributes?
You might think of flash fiction as a very short story (it is sometimes also referred to as a short short story). Most magazines that specialize solely in flash fiction don't accept fiction that is above, say, 1000 words or 1500 words, maybe as much as 2000 words at most. Flash Fiction Magazine and Flash Fiction Online both have a maximum word count of 1000 words, for example. By contrast, some (though not all) magazines that specialize in traditional short stories require a minimum word count of around 2000-3000 words. Some magazines also accept both flash fiction and more traditional short stories.
It differs from poetry in that poetry typically puts a stronger emphasis on the aesthetic qualities of language such as rhythm, rhyme, alliteration, metaphor, simile, etc to create layers of meaning on top of the literal text. Most (though not all) poetry is written in verse. That being said, there probably is a certain degree of overlap between what is known as a prose poem (a poem written in prose as opposed to verse) and a piece of flash fiction that uses very lyrical language.
The other forms that you've mentioned (twitterature, dribble, drabble, and sudden fiction) all sound like specialized forms of flash fiction. I know that the drabble, for example, is a concept that originates in fandom and is traditionally a story that is exactly 100 words, though some people don't use this strict of a definition for it.