`暖和` is almost exclusively for physical, ambient temperature (weather, rooms, your body).
`温暖 (wēnnuǎn)` can describe physical temperature, but it is also widely used for abstract, emotional warmth (love, kindness, comfort).
Correct: `他的微笑很温暖。` (His smile is very warm.)
Incorrect: `他的微笑很暖和。` (A smile cannot have an ambient temperature.)
Correct: `这个房间很暖和。` (This room is warm.)
Also Correct (Slightly more formal/literary): `这个房间很温暖。` (This room is warm.)
Rule of Thumb: If you're talking about feelings, family, a kind gesture, or a welcoming atmosphere, use `温暖`. If you're just talking about the physical temperature of the air or your body, `暖和` is usually the best and most common choice.