In public I'd just do whatever I'd do in private. Hugging, comforting, validating, empathizing, distracting. Often a removal from the scene helps. When ds was that age I'd throw him up on my back and get walking and he'd usually calm down pretty quickly (though throwing a tantruming toddler up on your back is a bit of an art form let me tell you!).
I think it depends on what is causing the tantrum too. Like if my toddler is getting restless, and then tantrummy because I'm expecting too much of him (for ex. waiting forever in line at the grocery store with tons of chocolate bars right next to him that I'm not allowing him to have), then I don't have a problem giving him a little something nice (I'm thinking a treat like a fruit leather or whatever) to distract him. I guess you want to do some thinking about "rewarding the behaviour" or whatever, but my take on it is in a situation like that when I'm expecting "big kid" behaviour out of my toddler it's my responsibility to help him succeed, and it's not really a fair expectation to begin with. So I should have tricks of distraction ready to prevent a tantrum, and if I forget (ex. get caught up with my shopping and forget to give the fruit leather before the check out line) then that's my fault, not his. Just my take on it anyway....
And about public tantrums don't forget that anyone there who has parented a toddler has had their share of tantrums. No one is judging. Or, well, if they are, then they shouldn't be. If you feel really uncomfortable (which I know I have when one of my kids was really freaking out), then I would remove myself to a more private place (outside, bathroom, car, etc). Often tantrums stem from being overtired, overstimulated or hungry, so it can be appropriate to cut your outing short to go home and rectify the situation.
Good luck mama, enjoy the wild ride that is toddler-hood!