Which is correct?
Both.
The first letter of the Hebrew word is the letter ches which makes a sound like the ch in the German word ach. (If you know IPA, it /x/).
Since there is no corresponding sound in the English language, the transliteration can be done either way.
Zev Steinhardt
Is one proper, and one informal?
Nope. Feel free to use either or some other variant.
Zev Steinhardt
Todah Rabah.
Chag Same’ach.
;j
As Zev stated both are correct. The CH is the anglicized spelling of the hebrew word (i.e. not using the hebrew alphabet but rather the alngo alphabet). The H is the anglicized pronounciation.
Don’t forget Hanukah, Hannukah, Hannukkah, Channukah, Chanukkah, and Channukkah.
You forgot Hhannukkahh and Chchannukkahh.
Not to mention Chachacha, and KHAAAAAAAAAAN-ukah. ;j
Happy chak- happy chag - happy holidays, everyone. 
(Not going to mention Quabalah. or Cabala or Kabbalah or…) ;j
I think I did a search of Historical Newspapers from 1920-1990 or so, one time.
Don’t know if I posted the results here or somewhere else. (or even if I did).
I’ll search for it.
o/ So drink your gin-and-tonic-ah, and smoke your marijuana-kah, if you really, really wanna-kah, have a happy, happy, happy, happy Hanukkah! o/
The eighth letter of the Hebrew alphabet is what’s in question. According to the Library of Congress Hebrew and Yiddish romanization (.pdf), it should be transliterated as an H with a dot under it. Since we don’t necessarily have the Unicode characters available for underdotted letters, I usually show it here as H.
Therefore: Hanukkah.
The Ch- spelling is based on German. Are you really sure you want to imitate the Germans? After what happened?
Spell Hanukkah with H for het. It’s the American way. If you disagree, you don’t have to get all het up about it.