As you are including Russia, EU would probably won. Both have considerable military forces, but the sum of all Europe militar forces would result in more soldiers. Then you got the biggest advantage of all, territory. The sum of the European continent with the asian part of Russia is massive and way bigger than NA. In a war like that you have to invade soon or later, and that extension is just unmanageable. Despite that, the cultural factor improves EU bets. Russians are a tough people, they survived and fought to death in 2 massive invasions. If NA passed the entire Europe and got in Russia, they would be debilitated and wouldn't be able to resist a harsh fight there.
In a war, the two most important factors is how big your country is and how many natural resources you have to keep fighting. Technology helps, but if you don't have oil and steel it is completely useless. And in this aspect EU would have a advantage. That's the single reason why Germany lost WWII. They were massively superior technologically, the problem is that it was they against the world, or the limited natural resources in german territory vs. all natural resources of the world. Their only chance would be win the war so fast that they wouldn't run out of resources, but the sheer size of the territories they would have to conquer was a big problem. And back to your question, the NA or EU technology is nowhere near the massive advantage Germany had. Those guys had vastly superior machines than the rest of the world.
And, back to reality, a war like that wouldn't have a winner. Just look at how expensive Vietnam was or Iraq was. Both would end completely economically broken and the occupation of the loser would cost a fortune. USA, Germany, France, UK would be nowhere near the top of the largest economies ranking and the BRIC (except Russia of course, since they would be in the conflict) countries would enjoy a lot their reign as biggest economies "helping" the broken NA and EU to recover (and profiting a lot in the process).
