As with all modal verbs, the past participle gemocht is used in the perfect tenses only when the verb is not followed by another infinitive: Käse habe ich noch nie gemocht. – “I've never liked cheese.” When an infinitive follows, mögen itself also stands in the infinitive (double infinitive construction): Ich habe sie nicht fragen mögen. – “I was hesitant to ask her.”
Möchte and its forms are grammatically the past subjunctive of mögen, but are never used as such in contemporary German. The conditional of mögen needs to be paraphrased with würde: Wenn ich Käse mögen würde, könnte ich viel mehr Gerichte essen. – “If I liked cheese, I could eat many more dishes.”
Möchte is in fact used as a somewhat politer synonym of wollen (“to want”) in the present indicative: Wenn ich Käse möchte, sage ich dir bescheid. – “If I want cheese, I'll let you know.” (Synonym to: Wenn ich Käse will,...) In colloquial German, the split between mögen and möchte is so clear-cut that the latter may be interpreted as an independent verb and hence used in the infinitive: Du hast hier gar nix zu möchten! – “You're not in the position to express wishes!” This is, however, nonstandard.
In dated or regional speech, the present indicative of mögen may be used interchangeably with möchte, particularly in questions: Magst du was trinken? – “Would you like to drink something?” This is normal in Austro-Bavarian areas. In most other regions it is less common and likely to sound affected.