Comparing a German payslip with an American one
The two systems divide the same money differently, which is why a comparison of headline tax rates misleads in both directions. German gross pay carries income tax, the solidarity surcharge where it still applies, church tax for members, and social insurance contributions covering pension, health, long-term care and unemployment. An American payslip carries federal income tax withholding, the state's own withholding where the state has one, and the federal payroll taxes, with health cover typically bought separately rather than withheld.
That last difference is the one most often missed. A German deduction that looks like tax is partly a purchase of health insurance; the equivalent American cost is usually a premium that never appears on the payslip at all. The net figures on this page compare what the payroll system withholds, so the German side has already paid for something the American side has not.
Why the two gross salaries differ
The salary shown for the second place is not the first salary converted at an exchange rate. It is the salary that buys the same basket there, computed from official purchasing power parities. Starting from equal purchasing power is what makes the comparison of what is left meaningful: because both sides begin with the same real income, the whole of the remaining difference is the tax and contribution systems.
Where one side is a US state, its price level is chained rather than measured: Eurostat publishes a parity for the United States as a whole, the Bureau of Economic Analysis publishes an index placing each state against the American average, and the state figure is the first scaled by the second. The page notes this above the tables. It is the standard construction and it still rests on one assumption more than a country-to-country comparison does.
The market exchange rate appears separately, and the two figures are usually far apart. That distance is the point. It is the reason a salary that looks like a large increase on paper can be a reduction in practice, and the reason the reverse happens too.