I've talked about complexity vs. complicated and simplicity before. It all depends on your point of view, and this Eric Berlow TED talk (3 minutes) highlights that beautifully. If you can take a different view on your situation, you can often take something that seems overwhelming and home in on the areas that can really influence the question at hand. In other words, there may be a simple solution hidden in the seemingly intractable.
There are many definitions of these terms out there, so you need to be careful when reading / talking to people to understand how they view the terms. In my mind, simplicity is the idea that there are one or two leverage points that will affect the entire system. That doesn't mean the system only has a few parts (that would be simple), it's just that those parts are all interconnected. Complexity, on the other hand, suggests that there isn't a small set of leverage points that affect the overall system. In this sense, I think Berlow's examples are about complicated systems, which are difficult to describe, but once you find a useful way to describe and visualize them, the simplicity can emerge. (And then there is a chaotic system in which it appears to behave differently for the same changes - the system is unpredictable, even if it is describable.)
There are some interesting comments regarding the talk on the TED page, if you want to see where people have gone with the discussion. Buddha. Military. The "real" problem in Afghanistan. etc.