A stock is traded on NASDAQ in USD. But I can also buy it on XETRA in EUR.
What are the advantages and disadvantages for a European to purchase on a US stock exchange? Let's say my broker let's me buy wherever and the fees are the same.
Liquidity on dual listed equities is rarely the same on both exchanges. More liquidity means you would typically get a better price assuming you execute the trades using the same order types. It's recommended to trade where the liquidity is greater unless your trading method benefits somehow from it being lower. It's important to remember that some ADRs (some European companies listed in US) have ADR fees which vary. USD/EUR transaction fees are low when using a decent broker but you're obviously participating in the currency risk.
No, there are neither advantages nor disadvantages. I'll take on this question from an accounting standpoint.
Now, if we're talking about comparing markets this is a entirely different question.
Should I buy stock of Company A, who is in the American market (as an European)?
Should I buy stock of Company B, who is in the European market (as an American)?
I would recommend this as additional level of diversification of your portfolio to inlcude possible large inflation of either the currency. The possible gains of this foreign exchange may be greater if one or the other currency becomes weak.