Hey Simon,
Comet Cache is a page caching plugin, which means that it captures the HTML output generated by WordPress and stores it in a static HTML file. If you have dynamic content on the page, such as a Twitter widget, that dynamic content must use JavaScript to do the updating (JavaScript is client-side and runs in your browser). If it’s using PHP to update the front-end, then it won’t be compatible with WordPress caching plugins like Comet Cache.
I recommend contacting the plugin developer to see if the plugin is compatible with WordPress caching. If it’s already using JavaScript to do the updating, then you might want to check the JavaScript console in your browser to see if there are some JavaScript errors that are preventing the plugin from updating the Twitter feed.