Why are there two different words? According to Merriam-Webster's Dictionary of English Usage, back in the 1920s the National Fire Protection Association urged people to start using the word 'flammable' instead of 'inflammable' (which is the original word) because they were concerned some people might think inflammable meant not-flammable. Actually, the in- in inflammable was derived from the Latin preposition meaning en- (like enflamed), not the Latin prefix meaning -un. It's not like everyone knew the derivation of the word, so the change probably made sense. However, confusion persists today regarding which word to use.
Flammable is the modern term for a material that catches fire readily. Inflammable means the same thing. If a material won't burn easily, you could say it is not flammable or nonflammable. I don't think unflammable is a word (and really anything can burn if you try hard enough, right?).