There is a very simple answer here - there is no difference in meaning. The spelling with 'e' is British, the spelling with 'i' is North American. The same goes for the nouns, 'inquiry' and 'enquiry'.

There are of course other differences in spelling between American and British English. The most common ones are words that end with 'our' in British English and are spelled 'or' in American English - labour (labor), honour (honor), and so on.

Another common difference is words that end in 're' in British English and are spelled with 'er' in American English- theatre, centre.

And finally, words that end with - 'ize' in American English and are often spelled with - 'ise' in British English - sympathise, criticise, and so on.

Luckily, my spell checker accepts both!

See also:
JustSomeLyrics
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徳永英明 まちぶせ Lyrics
vasco rossi 09VaBeSeProprioTeLoDevoDire Lyrics