They both have warm and sincere characters. Both drama have this family vibe, it's that type of drama that makes you feel warm inside! Both are really good dramas, definitely worth checking out!
Both are centered on a Japanese family and have a similar moods.
Both female leads suffer from social phobia. There are many scenes in Heart to Heart that are very similar to the scenes from Castaway on the Moon (the helmet and umbrella ones), the director probably got inspired by the movie.