The ultimate burden is the same, assuming equivalent activities. But under a consumption tax, the consumer (i.e., the ultimate taxpayer in either system) has more control over whether or not he actually pays tax. Apart from spending on bare necessities, he can sit on his money until he decides it's time to spend it. The government only gets his money when the conditions are such that he's willing to spend, which puts added incentive on government to keep taxes down.
Both the flat income tax and the sales tax are consumption taxes. I'm not sure what your point is.