Frankincense is an incense that has the word incense in its name, so it's pretty clear that there's a connection. Indeed, frankincense is a portmanteau of the French words franc (meaning "noble" literally and "high quality" figuratively) and encense (French incense). Franc in all likelihood is named after the Frankish tribes who lived in the area during the Middle Ages (since, I don't know, they became French nobles later?). Their Old High German self-appellation was franko, which probably has roots in the Germanic word frankon, or "javelin". Encense, on the other hand, takes a predictably more Italic route, going back to the super-fun Latin word incensum, "that which is burnt", from the verb incendere, "to set on fire" (and, as you can see in the Harry Potter infographic, the root of the spell incendio). Eliminate the prefix in-, and another verb, candere ("to glow") can be identified. This, through Proto-Italic kandeo, goes to PIE kand, also "to glow", but MOST IMPORTANTLY it is the etymon of candle, through later candela. So if you ever light an incense candle, you're glowing a glow, to be frank!
Adam Aleksic, a rising sophomore studying linguistics and government at Harvard University, has been described as the internet's sixth most famous etymologist. He also has disturbing interests in politics, vexillology, geography, board games, conlanging, and law.