The difference is not always inherently obvious. I am a person with an anxiety disorder - I constantly worry about things going bad and cannot help but imagine the worst possible outcomes. It's easy for me to confuse the two, because I see a risk and associate a bad outcome with it, which can lead me to thinking that it's a mistake. This is wrong.
A risk naturally involves the notion that one might be mistaken. If I was sure of an outcome there would be no risk and I'd not take a choice that would lead to a bad outcome. I spent a lot of the last decade working in financial services, where risk is an entirely different thing. (Let me explain to you some off-line time about tail risk, black swans, and why people who think they can control and predict such things are prime targets. See London Whale.) The core lesson in all of this is that (some) risk is inevitable. You cannot hedge, lay off, or otherwise avoid all risk.
If you are then always taking risks, the question is what's a mistake? Well, under- or overestimating your risk is a mistake, clearly. So is under- or overestimating your payoff. But we do this all the time (see "lotteries" and "day trading"). It seems clear that the way a risk or payoff is stated can have an outsized influence on peoples' behaviors. That's relevant here only in that I think we form internal models of risks and chances of payoffs and sometimes we make what others see as mistakes because in our (internal) predictive model we don't see it that way.
Even if we do see a risk we may do a risky thing: sometimes we take risks because we feel it's necessary, or because we feel we can control it, or because there's no realistic alternative (see "driving in non-urban America"). Sometimes we choose to avoid some risks that we might safely take or engage in a risky activity we might safely avoid.
Here the line around "mistake" gets blurrier. Different people estimate risks differently and are willing to accept different risks or risks of different magnitudes. In the case that brought this up last night I expressed concern that my friend might be putting themselves in a situation where they would get hurt. They assumed that I thought they were making a mistake.
In this case I wasn't make that judgment - I was worried about them taking this risk. The two are different, but it's sometimes not clear until afterward which is which. Other times we not only think we can, but we have to behave as if we were certain: "Taking that job would be a mistake" ... "it would be a huge mistake to start dating that person."
I'm not sure I have any profound insights in this area; maybe this is just a call for me to pay more careful attention.