On short training runs, you can get away with going outside for a bit and if you forgot the sunscreen, well, nobody minds. But, as you start training for half marathons, marathons and longer, sometimes you are out for longer than an hour and you start getting into the likelihood of getting burned. So, don't forget to wear sunscreen. And, make sure you choose a kind that will let you sweat and not fade away. Try one of the many sport varieties that are available. Because nothing is worse than getting back from a great run, only to find out you have a sun burn.
And one more thing, if you are putting the sunscreen on your arms or neck, don't forget that as you run, your clothes may shift a bit, so you might want to keep that in mind so you don't end up with a small stripe burn when your clothes shift. Take it from someone who has been burned like this, it is not very much fun!!