I downloaded this plugin and took a look and it does work as advertised for me. The function it uses for displaying the page list is:
Which is very straightforward and does very explicitly order the items via menu_order. However, because it's a core function subject to actions and filters, you might have another plugin (or maybe your theme) interfering with the ordering. Before getting carried away, first ensure that the Order is explicitly set on each page by editing (or quick editing) each page in question. If two pages are set to "0", they may order one way in the page list and another way in this widget.
If that checks out, I would check three more things in this order:
- Make sure you're using the latest version of this plugin.
- Make sure you're using the latest version of WordPress.
- Try disabling all other plugins
Let me know how you make out here!