Both by the same mangakas. They are drama series with lots of comedy elements and caricaturesque drawings. Both deals with deep human feelings, the world of dreams and the connection between all human souls and destiny.
The main male protagonist is a hikkikomori, and the story has some focus on his recovery with the help of the female protagonist(s). Welcome to the NHK is a more realistic story with a heavy focus on hikkikomori issues, whereas Rozen Maiden is a fantasy involving living dolls, but the male protags' struggles in both series revolve around their status as shut-ins.