First of all I would say that "Bedeutung" in German comes closest to "meaning" in meaning in general. E.g. the "meaning of a word" would be "die Bedeutung eines Wortes". Without context this is how one would in most cases understand the word.
Now, "Bedeutung" can also mean "significance" in the sense of "importance", but then you usually talk about "Bedeutung von etwas" ("significance of something"), e.g. of an event. If you then wanted to say that you were unfamiliar with the fact that said event had such a significance, you would rather say something like "Mir war unbekannt, dass dieses Ereignis eine solche Bedeutung hatte." in German. "Diese Bedeutung ist mir unbekannt." could have the same meaning, if everything was clear from context, but it would still sound very unnatural. Although I must say that the English sentence "This significance is unfamiliar to me." also seems unnatural in a similar way in my opinion.