Cucinare versus Cuocere?
Both seem to mean cook. The only difference I see is that cucinare is a regular verb and cuocere is irregular.
When is it appropriate to use each term, and how?
Cuocere means to subject something to heat, and as such it can be used to refer to inedible materials too: for instance "cotto" as a noun often refers to cooked clay (but it can be a shortening of "prosciutto cotto" as well). "To bake" is usually translated to "cuocere in forno" (cook in oven).
Cucinare (derived from cucina, kitchen, which derives from cuocere as well) refers to processing food in a kitchen, from preparing the ingredients to cooking to assembling the final product. So it does include cooking, but has a broader meaning: all the steps in a recipe, including cooking, are "cucinare".
On the other hand, cucinare isn't very used to refer to preparing cold food, even though cooking might be involved.
I think the interesting point should be that Duolingo is at cross purposes with the two verbs and should really be following along with the words that it has used previously before introducing a completely different word or accept the solution given. As f.formica says they both mean to cook and both should be acceptable answers.