It's my first time answering a question but I just spent a year studying the history of the English language so I feel like I can answer this.

First off, North American English (NAE) is what linguists would call a language variety rather than an accent. An accent would mean that Americans simply pronounce things differently. A variety better describes the way Americans speak differently, including things like different pronunciation, different words used (lexis), different grammar and different spelling (orthography).

NAE is also the name we give to the standard variety which influences the US, Canada, and places like the Phillipines, Liberia, Guam or American Samoa. The rest of the English-speaking world is influenced more by the British English (BE) standard variety, which leads you to group Britain, New Zealand and Australia into the "traditional English accent."

(In reality, of course, each of these places has many sub-varieties and even within the US there is great variety nowadays. Some different varieties are Mid-Atlantic, Northern, New York, Southern, or AAVE (African American Vernacular English).)

So, why are they different? The first important thing to recognize is that language continually changes, and some languages changing in some ways faster or more slowly leads to different varieties, simply put. Secondly, NAE is actually generally more conservative or "traditional" than BE! Many NAE features are ones that are preserved from 18th-century Britain. Many settlers that came from Britain also came from northern England, Scotland and Ireland, which were more conservative about changes that were happening in population centers like London in the 18th century.

Here are some differences between BE and NAE:

• Most NAE varieties are rhotic, which means they pronounce the "r" before consonants and at the end of words as in bar, barn or beer. BE started dropping these "r"s in the 18th century. I think this a good example of what people often percieve as traditional, but it's actually an innovation of the BE variety!
• Subjunctive is preserved in constructions with e.g. insist in NAE. A NAE speaker might say I insist that he leave and a BE speaker might say I insist that he leaves. The negative form, however, is a NAE innovation (I insist that he not leave).
• There are many lexical differences. An example is that NAE preserves diaper while BE uses a new word: nappy.
• Spelling differences are interesting as well. I'm sure everyone is aware of e.g. NAE color and BE colour. US English is most noticably different here, while e.g. Canada uses a mix of NAE and BE spelling norms. Spelling in the 18th century still had some variants, meaning you could often choose how you wanted to spell a word. Noah Webster wrote a dictionary in 1828 that was influential in establishing an emerging NAE norm and you might find some reasons for suggesting different spellings in his preface. Here's an excerpt:
  • "It is not only important, but, in a degree necessary, that the people of this country, should have an American Dictionary of the English Language; for, although the body of the language is the same as in England, and it is desirable to perpetuate that sameness, yet some differences must exist. Language is the expression of ideas; and if the people of one country cannot preserve an identity of ideas, they cannot retain an identity of language."

This was a comparison of a broad North American standard variety and a broad British standard variety, keep in mind that in most areas, there are many sub-varieties. There are also many more changes and differences I couldn't all stuff in here.

Most of this information is based on the following book:

Brinton, Laurel J., and Leslie K. Arnovick. The English Language: A Linguistic History. Oxford UP, 2017. pp. 443-8.

Might I also suggest this short basic video on the entire history of the English language which also contains some perspective on the NAE/BE differences and reasons for them :)