To put it simply, the differences are because of the lower quality mentions.
When you run a Quick Research, we also include lower quality mentions that are not filtered by the quality filter, filter that your project is setup by default to ON. Also, a Project imports automatically the mentions for the past 30 days.
To have the same exact results, you should start a Project with the Quality Filter set to OFF. Be sure to compare the results with the Quick Research results for the past 30 days.