There's something positively elusive about the distinction between the two. I know I have talked with fairly intelligent people, tried to convince them that there is a difference without succeeding.
The main difference is the difference between the known unknown and the unknown unknown. Risk is something you know; you at least know the probability distribution of the unknown. Thus, if you need to hedge bets, you can do so in a manner that very likely achieves your goals, whether you are risk averse, risk neutral, or risk taker.
Uncertainty is about not knowing the probabilities (and distributions thereof) of future things, nor potentially the magnitude of their effects. This means it is harder to create an intelligent strategy to satisfy your preferences. You cannot forecast that making bets such and such you will distribute the risks you have and minimize them accordingly.
As an example: some variables have a gaussian distribution. If you know the mean and variance, you can easily create portfolios that manage your risks (and make risks of really bad outcomes really, really unlikely - unlikely enough that they just cannot happen). If instead, your variable has a power law distribution, this does not hold, because power laws tend to have a long tail - you need to figure out other ways of managing your risks. These examples are still examples of risk, even if the latter makes things more difficult (as you cannot rule out extreme outcomes).
Lastly, what if you don't have any clue about the distribution of the variable? Some have suggested that we should then make an assumption, whether it be a bell curve or a flat distribution. The problem with this assumption is that we have a high probability of being wrong in our estimate, which means our risk management is very likely to produce analysis that has nothing to do with reality - predictions will be far off from the actual distribution of events.
The thoughts have, at least partially, been inspired by the book Black Swan, which I heartily recommend. It is possibly the best critiques of modern financing system I have read, written by a professional trader.