Actually, widgets have a more complex role in the rendering of pages than I'm letting on when I claim that wikitext is a superset of HTML and that widgets are an extension to HTML. When a tiddler is being rendered, everything in its wikitext is transformed into underlying widgets, including HTML that you've embedded in your wikitext. Once this widget tree has been built, TiddlyWiki can convert it back to the HTML that causes the page to display in your web browser. $widgets in pages aren't an extra feature added to wikitext, but a means of directly creating a widget in TiddlyWiki's underlying representation of the page, rather than allowing wikitext or HTML to be transformed into a widget by TiddlyWiki.

You can read more about this in the Widgets section of the TiddlyWikiDev wiki.

Custom Widgets are an additional complication, in that they are actually procedures (which generate wikitext widgets) underneath.

Unless you dig into the JavaScript code underlying TiddlyWiki, this is not a detail you need to concern yourself with.