It's somewhat technical, but if you're interested, the problem with flat taxes is that they are only flat when considering the numerical value of the money, which is less than useful.
Fundamentally, what really matters is how useful a person's money is to that person -- it's marginal utility. As one would expect, when going from poverty to great wealth, money has a diminishing value to a person. While two bums might get into a fight over $20, Bill Gates can lose $10,000 on the stock market and not bat an eyelash. A person who can barely afford to feed himself values his 10% far more than a person who bathes in champagne.
Focusing on the just the nuraber of dollars is fairly wothless, when what really matters is how much those dollars mean to a person. Consequently, a flat tax is a crappy idea, because it ignores the value of people's money.
If there's any tax that is fair economically-speaking, then it would be one that attempts to flatten MARGINAL UTILITY. In today's terms, that would be called a progressive tax, which is what we have .... and now you should have an idea of why such a tax is preferable.