An extension is something that is specific to the browser, and they are a bit different on each browser, but tend to be able to learn more about the overall state of the browser; they may be automatically added to pages, accessible separately from a page, etc.
Add-on is more of a generic term which is used to mean a lot of different things. What it actually means just depends on who is talking, but the mozilla definition is probably as good as any; it could be anything that adds functionality to your web browser, regardless of the context. Hier sind die Top 5 Add-ons für Firefox
Extensions tend to be automatic once installed. Plugins are instantiated in one of two ways: 1) by an <object> or <embed> tag in the HTML of a web page, or 2) because they are registered to be the handler for a mimetype which the browser doesn’t support.