I just did a quick search to see if it could expedite my settling on a concise answer. There are a lot of bad concepts about this question out there. First, be mindful that "similar" is not the same thing as "equal".
To me, "because of" is used when there is a direct causal link:
"The road is wet because of the rain".
"Owing to" and "due to" I feel connect an event to a reason for the event happening. I prefer "due to", so:
"The game was cancelled due to the rain".
I am making the argument that the road getting wet is simply what always happens when it rains. On the other hand, we could have played our game in the rain. The rain was simply the reason why we chose not to play.
"Lots of our glasses get broken because of the vibrations of freight trains vibrating them off the shelf".
"I decided to move all of our glasses into a box due to so many getting broken falling off the shelf".
Final comment. It's just my bias, but I associate "on account of" with a Western. "My Pa hired a bunch of cowboys with rifles on account of cattle rustlers stealing our herds". It means the same as "due to" or "owing to", but to me it has an "old fashioned" flavor to it.
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