Are you sure about that?
It's going to depend very much on where you are, how you're organized, what tax breaks you get, and how much money you make, but states generally have both a) many different kinds of tax: income tax, property tax, sales tax and b) a cumulative higher tax rate. A much higher proportion of my money personally goes to the state than it does to the feds.
For for example you might have:
State A: State income tax: 5% Property tax: 5% Sales tax: 5%
State B: State income tax: 6% Property tax: 6% Sales tax: 5%
Yes, everyone in both State A and B pay, let's say, a federal 7% income tax and payroll taxes. But the state taxes make a huge difference. State says to Wal-Corp - build a store here and you'll be exempt from property taxes and sales tax for 5 years, and we'll give you a special state income tax credit that reduces it to 3%. That's the kind of deal making we have going on with states, except the federal government doesn't intervene to stop it like the EU did here.
http://www.newsworks.org/index.php/local/brandywine-to-broad...
But most taxes in the US are Federal.
In Europe they are effectively 'state level' - making the problem worse for Europe than for state-by-state tax competition in the US.
Re: your Ireland comment - yes - I see your point, but the issue here is not Ireland having a 'lower tax rate' of 12.5% - rather than they allowed Apple to have 'special privileges' thereby paying next to nothing, right?