I'm posting this solution as its taken hours of fruitless web searching to work out what the problem was and how to fix it.
When you create a new sub domain site, always type the name in lowercase. If you use upper and lowercase, this alone can cause the visual editor toolbar (for posts and pages) to display incorrectly.
In short, I still saw the 'Visual' and 'HTML' tabs, but the buttons in the 'Visual' tab were simply the HTML buttons.
If its clearly that important, then perhaps Wordpress should automatically substitute uppercase characters to lowercase ones to avoid stupid users (errrmm, me!) from being able to make such a mistake.
Might submit that to Trac...