A holiday is a special day for religious or cultural reasons - such as Christmas, Easter, New Year's Eve, and your country's Independence Day. Some holidays are also official days off from work, and others are not.
A vacation is when you take multiple days off from work/school to relax and/or travel. However, American English speakers call this a vacation and British English speakers call this a holiday:
My parents are on vacation in Asia. (American English)
My parents are on holiday in Asia. (British English)