Some synonyms for "unwittingly" would be "ignorant of", "unaware of" or "uninformed about". You could "unwittingly" offend someone if you did not know the customs of their country. For example, I "unwittingly" made my Japanese daughter-in-law uncomfortable by highly praising her cooking. I didn't realize that such high praise made her uncomfortable. Some synonyms for "inadvertently" would be "careless", "uncaring" or "thoughtless". For example, when my son poured the chocolate milk from the jug, he "inadvertently" spilt it on my white tablecloth. He should have been more careful! He knew I wanted to keep that tablecloth clean. He should have poured the milk into his glass when he was in the kitchen. Both words can mean "unintentional", but they imply additional reasons for the unintended actions. Hope this helps.

They mean the same thing, although unwittingly implies a lack of awareness more than inadvertently does. Really though this doesn't matter and both words will make sense. Here is an example where, personally, i think unwittingly sounds better: EG: "He unwittingly walked onto the wrong train" instead of "He inadvertently walked onto the wrong train" In this context unwittingly sounds a little bit better to me, maybe because the situation is less severe and walking onto the wrong train is something you do out of a total lack of awareness like if you're daydreaming or something. Here is an example where personally i think inadvertently sounds better: EG: "He inadvertently started the war" instead of "He unwittingly started the war" Inadvertently seems to sound better to me here because of the seriousness of the act and the fact that there's more of a possibility for some awareness over the potential for an accident to occur. In other words, he didn't mean for it to happen but he knew that it could happen.

The two words are synonymous (have the same meaning) and are interchangeable. They nothing mean unintentionally...

I think Reece is correct that “unwittingly” implies unawareness. That implication is an important difference between the words. Don’t use “unwittingly” if the person in question was immediately aware of his mistake/action. Example: He inadvertently dropped his keys, but immediately picked them up. Compare to: He unwittingly dropped his keys on the train, and when he finally realized his mistake (hours later after getting off the train) it was too late to find the keys.