None of them are incorrect. A "spoonful" means that you've got enough to fill a spoon, while a "cup of" means that you've got enough to fill a cup. The word you use depends on the situation. For example... None of them are incorrect. "Could you get me a cup of tea?" "Could you add a spoonful of sugar to my tea?"

Grammatically both are correct. You usually see these expressions in cooking recipes, when they report how much of each ingredient you need. The meaning of the two expressions is different, because the denote different quantities. If a recipe asks for "a cup of sugar", you need as much sugar as a cup would hold. If it asks for "a spoonful of sugar", you need as much sugar as a spoon can hold. You use them in sentences when you talk about quantity, e.g.: "A spoonful of sugar is enough to sweeten a whole pot of tea" "A cup of sugar isn't enough for this cake. You need two cups." Spoonful certainly is used in British English. I am not sure about American English but I expect it would be understood, at the very least. Finally, you spell "sugar" with an "a", not an "e" 😊

Because if you want to talk about how many cups you have of something then you can say you have to use a tablespoon of milk, 3 thirds a cup of chocolate or two large eggs and you have to whisk them all in the bowl or a quarter cup of butter or something like that

A spoonful is most likely British because we don't use the same words to say anything at all besides the normal everyday phrases to start a conversation

The first one is American English and we use a cup of sugar when we cook bake or measure something