Not a big difference meaning wise. It is just the way you use them… As per Judy’s answer . You can also I magine 2 people have just had an argument, you shut the door. After a while you hear a knock. You know it is the person you had argued with and you open the door agitated and say “ what else”?

First it would not be give me a sugar. You could say the sugar or pass me the sugar or I would like some sugar. :What else" goes at the beginning of the sentence. What else would you like? " Anything else" would go at the end. Would you like anything else? You can also use something else. "Would you like something else?"

Not much difference, but perhaps anything else sounds just a bit more polite. It all depends on the tone. Hope this helps :)

What else can sometimes sound aggressive if used wrong so maybe stick with anything else until you get the connotation （ニュアンス） down.

"What else" presupposes that you want something additional, and the question is just to find out what. "Anything else" is a yes / no question, where the person asking doesn't know whether or not you want anything. We can say "a sugar" if we mean a cube or a pack of sugar (this presupposes that the restaurant has sugar in cubes or packets), but otherwise "sugar" is not usually countable.