The EU is a monetary union, not a free-trade area, like NAFTA, however there is free trade in the EU.
Free trade seeks to utilize the theory of comparative advantage to achieve mutual benefit from those involved. The theory of comparative advantage says that through specialisation and trade, countries can produce outside their productive capacity and PPC (production possibility curve) and produce up to their TPC (trading possibility curve). Theoretically, then, where there is comparative advantage, there is a potential for mutual benefit from trading. The theory of comparative advantage doesn't, however, take into account transport costs or any trade barriers such as tariffs, so in order to fully benefit from comparative advantage, there needs to be a free trade agreement such as the one in the EU, which suggests that the EU benefits from these advantages.
There are, however, drawbacks. Firstly, the initiation of a free trade agreement, while it will create new trade partners (trade creation), can harm trade with countries outside the free trade area (trade diversion). Secondly, international trade leads to higher levels of competition in the market, and can therefore "kill off" under-performing industries, as well as infant industries. This could have been part of the problem for Spain in 2009, as some of their industries may have been functioning well at a domestic level, but after the increased competition entered from overseas, such industries could no longer remain competitive and were forced to close. This would lead to higher reliance on imports for the Spanish economy, and could also lead to a long-term lack of competitiveness as infant industries that would grow and become competitive lose the help they would normally get from the government in the early stages of development.
A further problem of free trade in areas such as Europe is that if one economy enters a recession, it can affect the entire free trade area, especially if the share a currency such as the euro. This is because as soon as one country in the eurozone starts to underperform, foreign investors will lose confidence in the whole area and stop investing which would lead to a lack of FDI and limit AS. As this starts to happen, confidence within the trade area starts to decline and less products get traded as well as businesses losing confidence and not investing in development and research. Such confidence problems can quickly spread throughout economies that are so closely integrated. The quote on p11 of the stimulus from a speech by Pascal Lamy illustrates that "Internatonal trade was a casualty of the global recession and world trade shrank by 12% in 2009." Which supports my point.
To conlcude, while free trade is useful in increasing the productive capacity of all economies in the free trade area, it can be dangerous to integrate too closely with other economies as it is easy for one failing country to drag all of the others down.