One difference I see is that "The plan came into being" is in active voice while "the plan was brought into being" is in passive voice. I will still leave this to experts to answer, though.
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voynichThe plan came into beingThis almost sounds like the plan materialized by itself from nothing, as if by magic. The reader will take the meaning correctly, though, if you provide the right context.
voynichthe plan was brought into beingThis sounds like someone or some group made the plan come into being.
I googled "The plan came into being" and got over 100000 hits. Then, I judged that the sentence was a natural English sentence("The plan was brought into being" got only 3 hits though). How do I think it of?
Neither of them sounds natural to me.
voynichThe plan came into being ... over 100000 hits ... The plan was brought into being ... only 3 hitsI find this astonishing. I expected a difference, but not such a dramatic one.