Hi! I have been lurking around here and found a great amount of helpful info, so for once I wanted to add something from me.

In the Ads Manager you can see and analyze the effectiveness of ads in campaigns conducted in Facebook applications and services. Attribution on Facebook allows you to measure the effectiveness of conversions as a result of ads published in Facebook applications and services and with various publishers. If you notice discrepancies between the reports in Facebook Attribution and Ads Manager, we recommend checking the default settings first:

Date ranges: Check that the reporting period in Facebook Attribution is the same as the date range in Ads Manager.

Attribution ranges: Check that the same attribution range is selected in both Facebook Attribution and Ads Manager. The Ads Manager offers only 3 attribution ranges, while the Facebook Attribution offers more options.

Attribution models: Check which model of attribution you use in Facebook
Attribution. Note that the Ad Manager uses the attribution model of the last point of contact.

Conversion events: In Facebook Attribution you can select a conversion event to measure in the attribution reports. Make sure that the conversion you are interested in coincides with the conversion event visible in the Results column in the Ad Manager.

Hope it helps too!