Both 温かい and 暖かい mean the temperature being relatively high.
The difference is,
温かい means the warmth physically measured.
暖かい means the warmth as subjective feeling.
This is describing other persons face. So it is not proper to use 暖かい, because you don't know how others feel.
This is describing the feeling that one subjectively felt his/her 胸 and 手 got a little warmer. So it is proper to use 暖かい.
温かい, 熱い and 冷たい belong to the same group, as physical temperature.
暖かい, 暑い and 寒い belong to another group, as subjective feeling for temperature.
※I modified the explanations of 1 and 2 for easier understanding.