To put it simply, the differences are because of the sources of the mentions, their quality and the project settings you've made.
The settings within the Project category are highly customizable. You can fine-tune your analysis according to your exact needs and the mentions will be reported according to those settings. Within the Quick Research Tool, the settings are more limited, and only a few parameters can be customizable. This is why, most likely, you'll get to see differences in mentions between the two sections.