Seems to be the typical British miscommunication of "public" and "private" here.
What do you mean by "public money"? The banks the government bailed out a decade ago were private entities full of private money. They're not government institutions. Bank of America is not a government bank, it's just named Bank of America.
So there is literally no difference between the government giving bailouts to private banks or other private businesses. The point is that if its okay for the government to bail out one private business, shouldn't it be okay for them to bail out another?