Actually, 'inquiry' and 'enquiry' mean the same — it is a process of asking for details, information. They can also mean an official investigation.
We see the only difference between 'inquiry' and 'enquiry' — the first one starts with "E"-letter and is more common for Britain, while the second one begins with 'I'-letter and is more common for the USA.
'Inquiry' and 'enquiry' are pronounced the same way — [ɪnˈkwaɪə.ri]