"Möchten" is "would like" "Wollen" is "want" "Möchten" is a polite way to "want" something while "Wollen" is a more casual usage. It works the same for English. If you worked as a waiter, you would say "Möchten Sie etwas?" Instead of the less polite in this case "Wollen Sie etwas?" Its the difference between saying "would you like something?" and "do you want something?" It is a politeness issue and "place/manner" come into account here. I hope that helped :)
Not a native German speaker or anything, but my impression is that it's the difference between "wanting" and "wanting that." If I mean to say that you want something, like an ice cream cone or a glass of water, then I should use "moechten." If I mean that you want something to be the case, or if the object of your desire includes a verb (e.g., "I wish that he'd do x"), then use "wollen."
"Wollen" is related to willing or wishing in English. You will that, or wish that. You don't will or wish a thing. (You can wish for something; not sure if "wollen" supports this kind of use.)