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Again they are both calculating different items. One is purchases the other "sales" as defined by the retailer.
According to DG at AE, Amazon includes Kindle Firsts, Kindle Samsung, KDP Select monthly giveaways and KU as "sales" all of which are free to the reader.
Whereas Nielsen records what a reader purchased from a retailer with hard cash, a customer walked up to a cashier and plunked down $20 for their book. Nielsen doesn't count free stuff even if the author gets paid regardless if it has an ISBN number or not.
Neither report is right or wrong. They just are using different data available to them.
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