Both about magical girls that were both made in the 90's they are very similiar to eachor due to the fact guys fight over both of the leading girls ,and they are amateurs at using magic also they have have mean girl characters that consider them rivals in love. What the big differences are is that Akazukin Chacha is more a light hearted, slap stick, fantasy adventure, story and Hime-Chan is more of a school, magic, love story.
Both have a blonde female main character who transforms into a superhero magical girl in order to fight the bad guys, both female main characters are very silly, goofy, and funny. Both animes also deal with romance, and they both aired in the 90's.
ACC has great characters too like Azumanga Daioh and this anime actually has way more characters but still almost every single one of them is still memorable and likeable. This anime's sense of humor is different to azumanga but I think azumanga fans will get it.