포근하다 and 아늑하다 are similar in meaning. Definitions in the dictionary are "cozy or snug". I think we usually say 포근하다 when we feel relaxed and comfortable with blankets, pillows, sofa, etc. Someone could also say 포근하다 when she gets hugged by her boy friend or father. 아늑하다 can usually be said to describe some closed space like a room, house, tent, etc. It means cozy. 따뜻하다 is just "warm". You can say this word whenever or wherever you want. There would be rare possibility to make a mistake using this word. You can surely say your boy friend's hug feels warm.