Some teachers have taught that holiday is the British English word and that vacation is the American English word. This is not true. Both sides of the Atlantic use both words. The words holiday, vacation and leave are synonyms. They are often used interchangeably, but have a slightly different meaning.
A holiday is a day that is a celebration of something special. Sometimes it is also a day on which one does not have to go to work or go to school. The Old English is meaning holy day. It is a compound of halig (meaning holy) and (meaning day). Christmas, Easter and the Buddha's birthday are all religious holidays. The Queen's Birthday, National Day and the Hong Kong SAR Establishment Day are all secular (non-religious) holidays. Saint Patrick's Day (March 17th), the Seven Sister's Festival (seventh day of the seventh lunar month) and Confucius' Birthday are all holidays, but we do not get a day off from work or school.
The word vacation is from the Latin vacationem (nominative vacatio). It means time-off from work, but without a cultural or religious reason. It became a more common word over the last hundred years. In earlier times most people did not get this luxury. This word is usually not used for religious or cultural occasions.
Native speakers tend to confuse these two words, especially in the UK. People use the English language any way they want and tend not to worry about it being correct or not. So in the UK people use summer holiday instead of summer vacation at a rate of 3:1. In the US, it is the other way around at a rate of 10:1. If someone is going to misuse these two words, in British English they might use holiday to describe a vacation and the other way around in the US. However both words are used in both varieties of English. Both countries usually use holiday to describe Christmas and use vacation to describe non-cultural time-off work. In the UK, it's often a class thing; university students call their time-off vacations, especially the summer break. Time-off that everyone has from school or work are called holidays because they were originally mostly on holy days.
Another synonym for holiday and vacation worth mentioning is leave. It is a period of time when one has permission to be away from one's job. It is particularly common in the UK armed forces, where leave completely replaces vacation and holiday. It comes from the Old English word for permission, leafe. It is often used in the forms: sick leave, maternity leave.... The person has permission to not be at the workplace. However they are still not absent from work. In theory a boss could ask them to send an email or read some documents while they are in bed recovering, however unproductive that may be in practice. The word leave strictly means permission. Example: May I have your leave to try? However this use is formal and old fashioned.
(The other word leave, comes from the Old English lafan meaning to bequeath / to let remain.)
Yes, British English does use the synonym holiday more often than vacation. However both words are used by all varieties of English. Native speakers are usually careless about which word they choose to use and that is the nature of English.