"These days, the two terms are often used interchangeably. However, there is a difference between the two. Enquiry means to ask a question, and inquiry is a formal investigation. Yet another difference lies in the etymological source of the prefixes ‘en’ and ‘in’. ‘En’ comes from French, and ‘in’ from Latin. Inquiry has a formal and official ring to it, while enquiry is informal in its connotation.
In general parlance, it is understood that enquire is to be used for ‘asking’, while inquire is what constitutes ‘making a formal investigation’. In reality though, enquiry is preferred in British English, whereas the Americans are more comfortable with inquiry. As a matter of fact, it is only in British English that any attention is paid to the distinction. In US and Australian English, inquiry has, for all practical purposes, taken over."
As a side note, I did notice that Word and also my Blog spell check do not recognize 'enquiry' as being spelled correctly.