The connection is blaze, burn, flame. In most languages - Latin, Greek, Sanskrit - the root moved to describe the shining flame, or a flash. But in the Germanic languages it came to refer to the soot or burned material.
BTW, I crossed out the word inflammable because it should never be used anymore. Just get rid of it at every turn. It is confusing and ambiguous. It was the original word for something that can go up in flames, and is related to inflame, inflammation. But while the prefix in- can be used in that way in English as in inundate, insert, induce, it is more commonly used as a negator: inept, inoperable, inappropriate. So everyone got confused. "Wait. Does it mean that it can catch fire, or that it can't? It's kinda important." Insurance companies in the 1950's started demanding that their clients not use it at all. Flammable and Non-flammable are clear. Unflammable never caught on.
This was all an intro for an upcoming post on signal-to-noise in language.