Same pronunciation, same meaning. They are the same word, in effect. British English uses both spellings, American English tends to prefer 'inquiry'.

Either spelling can be used, but many people prefer enquire and enquiry for the general sense of “ask”, and inquire and inquiry for a formal investigation: I enquired where he was from The first enquiry in my inbox today was about when we open. We are going to inquire into the incident as the police did not treat us fairly. The lawyers asked when the inquiry will be completed. In practice, enquire and enquiry are more common and are used for everything in British English although we have started using Inquiry here for formal things. In the USA its Inquiry for everything. Interestingly my spell checker does not recognise "enquire" or "enquiry" at all.