禧 is more formal/literary than 喜 (joy/wedding). It appears mostly in set phrases like 新禧 (New Year's blessing) and 福禄寿禧 (fortune, prosperity, longevity, joy).
history
慈禧太后 (Empress Dowager Cixi, 1835-1908), the de facto ruler of late Qing China, used this character in her name, meaning 'compassion and joy'.
礻 is the left-side form of 示, an altar where offerings were made to spirits. It marks characters about ritual, blessing, and the supernatural: 神 god, 福 fortune, 祝 wish, 祖 ancestor. In 禧 it anchors the meaning of blessing bestowed from above.
喜 supplies the sound xǐ unchanged and doubles as semantic, pure happiness, the same joy on a wedding 双喜 banner. 禧 thus reads almost transparently: ritual joy, altar-blessing, the kind of happiness invoked in 新禧 New Year greetings and 千禧 millennium.
