祝 is a short form of 祝贺

祝,You can understand :①Said the good wishes of people and things。For example:祝福（blessing）庆祝（celebration） and（和）祝你身体健康（Wish you have good health.） ② 是（is） 中国（china） 的 姓（surname） 贺，With a gift to celebration . Don't konw so that you can understand.

zhu is commonly used as Verb and using in the initial of a whole sentence. I would recommend you not definate it as "congratulate" but "wish or hope", for expample: zhu ni sheng ri kuai le! (hapaay birthsday to you!) zhu ni cheng gong!(Wish you success!) zhu he can be considered as both wish and congratulate, no difference with zhu. ie: zhu he ni sheng ri kuai le! zhu he ni cheng gong! The key difference between those two words is zhu he can be used as N. ie: re lie de zhu he (Warmly Congratulation! )

祝贺-congratulate 祝-wish, ie:祝你生日快乐