It's similar to 'like' vs. 'would like' in English. (In fact the grammatical relation is the same: 'möchte' is the Konjunktiv II form of 'mögen', so there's no indicative '*möchten'.) 'Mögen' usually expresses a preference or emotion: - Ich mag dich gern. - Ich mag lieber Fleisch als Fisch. 'Möchte' is rather used for a desire, e.g. when asking for something: - Ich möchte bitte ein Glas Wasser. - Ich möchte nächstes Jahr nach Spanien fahren.

Formally, "ich möchte" etc. is the Konjunktiv form of "mögen" (to like), but it has become independent to some degree and acquired the meaning "I would like = I want". This is also the reason there is no infinitive form of "möchte". So: Ich mag Eis = I like icecream, it tastes nice. Ich möchte ein Eis = I would like an icecream, please give me one.