Basically, progressive taxation only works if people have stable incomes. Take two people, one makes $30,000 per year, one makes $60,000 every other year. At the end of every two years, the two will have the same net income, but former will have been taxed (Federal income tax) $8,165 while the latter was taxed $11,187 because he looked richer than he really is. Ordinarily, this isn't a problem--jobs are fairly stable and my example was contrived--but in a deep recession, people will find themselves with unemployment and juicy severance packages one year, but almost nothing the next, and they'll pay more taxes than they ever have at a time when they can afford it the least.
This isn't the first time wealth redistribution programs have backfired. As cutoffs for the Alternative Minimum Tax crept effectively lower, people not ordinarily considered rich started to be affected. At the end of the dot-com boom of the late '90's, many in the Silicon Valley middle class found themselves rich on paper. Through a few bad trades and a tumbling NASDAQ, some not only saw their money disappear, but a high tax bill because the money they never really had made them technically rich.
None of this happens with a flat tax. Every dollar made is taxed the same, regardless of when it was made or how many dollars came before it.