Here is an OK picture kind of showing the difference.
http://upload.wikimedia.org/wikipedi.../B6-v6_180.png
In the boxer engine the opposite piston pair moves in opposite directions. So the pair is either moving towards or away from each other. This helps balance out the movement of the pair of pistons. In a flat V setup the opposite pistons move together (both right or left at the same time). This would normally create a very bad imbalance and is why you would never see this setup in say an opposed twin engine. However, since the V12 is basically 2 I6 motors the left bank and right bank are each balanced so the switch from I6 to V12 doesn't affect the balance of the system.
A boxer V12 would be possible but it would add a bit of length and extra machining cost to the motor. The machining cost would be because you have more crank journals to machine. The length would be because you simply can't pack the connecting rods as tightly if you don't share the crank journal. I believe all of Porsche's opposed 12s were also flat V12s rather than boxer 12s.