Both stories start off basically the same.
The main girls (Nanami in Kamisama Hajimemashita and Yume in Momo) are also quite similar. They are both kindhearted girls who got abandoned by their irresponsible fathers but who don’t give up their hope and try to somehow get alone with their situation. They meet someone who changes their lives (Nanami meets Mikage and Yume meets Momo) and turns them from an ordinary girl into something extraordinary (Nanami into a land god and Yume into the earth’s representative). At the beginning they’re both a bit gullible about their new role and have to prove themselves to others who think that they’re not suited for the job or don’t fit the image. Through their new role they also meet some possible love interests.
In the end, I enjoyed both mangas and I had the same feeling while reading them even through there’re also many differences.
While Kamisama Hajimemashita has a lot of Japanese folklore in it, Momo has its very own version of devils which have little to do with the devils from the myths. But all in all I think that if you liked one, you should at least check out the other.