Yes there are a handful of reasons that using this plugin is more powerful than just using the system in place on the Pagespeed website.
The first thing to know, is that unlike the tool directly from Google which can only provide a report on a single page at a time, this plugin will check all of your WordPress URLs. Having access to all of the page reports at once lets us get some valuable data.
The "Report Summary" view is one area where we take advantage of having access to all of the report data at once. It takes all reports from your site and gives you information about your best and worst URLS, and your biggest areas for improvement. This is especially helpful on sites that have many different content types and layouts. These summary reports can also be filtered by URL type such as Posts, Pages, or Custom Post Types.
Once you get into the premium version of the plugin, you are also able to take snapshots of your report summaries, and you can compare snapshots with each other. This is great for testing out the benefits or consequences of different WordPress Plugins. Or for benchmarking your pagespeed improvements as you improve a site, for your own knowledge or to show a client.
The premium version also has scheduled re-checking of reports, as well as the ability to import URLs from outside of your WordPress installation.
This last reason is perhaps a bit bias, but I like how I visualized the data better in my version as well :)
You can find more information about all of these features, including short video demos about them at my website: http://mattkeys.me/products/google-pagespeed-insights/