Thousands of miles and hundreds of years.
Maize is the name for any of the countless varieties of the tall grass zea mays among English speakers who speak some form of the Queen's English. Corn is the name for maize among North American speakers of English.
The late Alan Davidson, author of The Oxford Companion to Food, says that the word corn is used in North America to refer to sweet corn that is suitable for human consumption. We have never heard any Midwest farmers referring to "feed maize" growing in the fields for their cattle; but many times we have heard them refer to "feed corn" as opposed to "sweet corn." We don't think the term maize is regularly used by English speakers on this continent.
Elsewhere in the world, the name corn is infrequently used, even beyond English. Corn is maís in Spanish, maïs in French, mais in Italian, and Mais in German (we see a pattern developing). Yet the word corn is often preferred over maize for food products derived from the plant: corn flour/corn starch, corn meal, corn syrup, corn oil (although some of these can be hard to find in many parts of the world).