I'm very confused by the existence of these apparently antonymous words, which actually mean the same thing. Which word should I use? Can both words be used interchangeably?
Both words mean the same thing, i.e. that something can be set on fire. The reason for the confusion comes from people thinking that the prefix in- of inflammable is the Latin negative prefix in- (which is commonly used in English, e.g. indecent). In actual fact, in this case it is derived from the Latin preposition in. It's easier to think about it with the word inflame. If you can inflame something, it is inflammable (inflame-able).
In most cases, it is better to just use flammable to avoid confusion and accidents.
The Free Dictionary advises using only flammable to give warnings:
Usage Note: Historically, flammable and inflammable mean the same thing. However, the presence of the prefix in- has misled many people into assuming that inflammable means "not flammable" or "noncombustible." The prefix in- in inflammable is not, however, the Latin negative prefix in-, which is related to the English un- and appears in such words as indecent and inglorious. Rather, this in- is an intensive prefix derived from the Latin preposition in. This prefix also appears in the word enflame. But many people are not aware of this derivation, and for clarity's sake it is advisable to use only flammable to give warnings.
Both are correct in all situations.
Same with regardless and irregardless (although the latter is more informal, and often declared incorrect: http://en.wikipedia.org/wiki/Irregardless)