Both dramas revolved around people of Satsuma Domain during Bakumatsu period, with mostly similar characters but written from different perspectives. Some actors in Atsu-hime also appeared in Sego-don but with different roles.
Both are Taiga dramas that focus on the life of a prominent female in the Shogunate - who both have extremely strong and straightforward personalities.
It also revolves around the shogunate. It's more detailed about what was going on in Japan politically at the time but it also gives you a breathtaking peek into life in the Shogun's harem.