Within aWP, these tags mean the same thing: you may use the interchangeably and randomly select which tag you use at any given time. However, WordPress sees them differently. WordPress uses the <!--more--> and <!--nextpage--> tags, but does not recognize the <!--newpage--> tag, so if you want WordPress to completely ignore the tags after aWP is deactivated, use the <!--newpage--> tag.
The <!--more--> tag signals to WordPress to split the post only once and only on pages with multiple posts.
The <!--nextpage--> tag signals to WordPress to split the post as many times as needed, but only on single post pages. WordPress almost treats these parts of the post as separate posts, and, in most themes, adds links to the single.php or page.php to navigate these parts when needed.
If you paginate your single post pages or your WordPress “pages” and you want the posts to degrade gracefully into the breaks you originally intended if aWP is uninstalled, use <!--nextpage-->, so the individual page links remain the same.