The capitalization of this holiday isn't the grammar/writing issue. It's the spelling!
Of course, like any other holiday, both words are capitalized:
Or if written with the informal name: July 4th.
Spelling independence is a little more of an issue.
You may find long lists of rules for ent/ence vs ant/ance if you begin a study on this--many of which are so confusing and detailed (emphasis on the fourth syllable from the right, use ent!! okay..that's stretching it...) that an average person cannot decipher them much less memorize them.
When rules are too detailed and confusing, a writer is better off memorizing spellings or using spell check.
I will leave you with two simple rules for ent/ence that I think are actually helpful:
1. Use ent/ence (not ant/ance) if the root you are adding it to ends with c or g:
This rule actually makes sense because if you used ant/ance, the c would then say kuh (innocant--innokant) and the g would then say juh (diligant--dili-gant).
2. Use ent/ence if the root you are adding it to ends with d:
Hope this helps--and hope you had a happy July 4th!