Are "all on a sudden" and "suddenly" similar in term of meaning?

I think the phrase you are looking for is "all of a sudden," not "all on a sudden."

This phrase has the exact same meaning as "suddenly."

The word sudden is an adjective that means happening or coming unexpectedly.  "A sudden drop in cabin pressure," or "A sudden thunderstorm."  Suddenly is the adverb of that adjective.  "Suddenly, the cabin pressure dropped," or "Suddenly, a thunderstorm came up out of nowhere."

To insert the phrase "all of a sudden" is more of a vernacular use or an idiom of the word "suddenly."  It is more figurative and therefore less formal.  Obviously in writing, it is muddier - but when spoken, it sounds more conversational and familiar.  Americans tend to use "suddenly" in writing as it is more formal and "all of a sudden" in speech for the opposite reason.