This is obviously a massive topic, but from my vantage as a foreigner living in Europe is that Europe already has (and pays for) all features of a federal government with two exceptions.
* Responsibility to implement treaties remains with the individual members.
* No ability to tax or set fiscal policy.
The advantage of a single set of laws is obvious. For starters I wouldn't be spending my Sunday reading a 500 page document on minor items describing how Denmark's implementation of an EU treaty differs from the treaty's wording. (Although it would cost the jobs of the people who produce such documents, and as such would fly in the face of the one central pillar of paneuropean identity).
The ability to tax and set fiscal policy is needed to offset the imbalance created by the single currency. Another lengthy topic, but basically the assumption that every EU member should "export" is simply impossible. A normal currency system would correct a trade imbalance automatically. The euro systematically undervalues the cost of German output buy forcing their European neighbors to transact in the same currency.
And then you have mafias, bankers, tax havens, uneven social nets, and a raft of other social problems that can only effectively be dealt with by a central government. The irony should not be lost that the American government has had better success battling the Italian mafia than the Italians have.