Enquiry and Inquiry are basically the same word, they are spelt almost identically and both mean to seek information by asking a question, or they can mean an investigation into something. However though, there is a subtle difference in how the words are used in British English.
When asking a question, in US English Inquire or Inquiry are most commonly used. In British English Enquire or Enquiry are more popular although slowly, as with many words and phrases in British English, the US preference of Inquiry is starting to take influence.
The subtle meaning difference in British English with Enquiry and Inquiry occurs when you have a formal investigation into something - In this situation Inquiry is used.
Enquiry is used in British English to make an informal query/ask a question and is very commonly used.
‘The Inquiry into the accident will start next week’.
‘The government is going to set up a judicial inquiry into the corruption scandal’
‘Excuse me, may I enquire at what time the concert starts?’
‘Good morning, I’d like to enquire when the next train leaves for London Paddington?’
‘Any enquiry about the opening hours please call the following number.’
If you’re in doubt what to use then use ‘Inquiry’ or ‘Inquire’, it’s how it’s used in the US and is universally understood.