Well, the big difference is that the US have explicit transfer payments between states, but Europe originally did not envisage transfer payments between (sovereign) countries.

Now, of course, OCA (optimal currency area) theory predicts that we need 0. different currencies, or otherwise 1. homogeneity (ie similar response to external shocks) - not the case in Europe, or 2. perfectly free factor movement (where factors are basically capital and labour) - not the case in Europe (mostly due to language), or 3. transfer payments.

Given that 1, 2 are not the case, Europe, by entering the EUR (thus foregoing 0) has basically committed to 3 - transfer payments. So, yeah. The rich EU countries now must pay for the poorer ones - that's the long-term consequence of the currency union.