It's important to note that there's no such thing as a truly "objective" position in Google, or any search engine. You should always expect some discrepancy due to factors such as: 

Our research has shown that search listings after the first page of Google tend to fluctuate a lot more often than those on the front page. 

So while discrepancies found for the top 5 results should be very small, results at #10-#20 are a lot more volatile and can easily differ by 5 positions each update.

Google search results are very dynamic. Most likely, anywhere from a few hours to a few days will pass in between the time Ahrefs pulls rankings and the time you manually check the SERPs.

These positions are likely to have experienced some fluctuations by the time you check them, especially if they were outside of the top 5 results.

This includes factors such as:

You'll find that if you ask some friends from different parts of town to check on your ranking position at any given time, they'll often report different numbers. This is because of how Google "personalises" search results by location.
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Likewise, Ahrefs is unable to pull search results from the same IP network as you. This accounts for some discrepancy - again, small in the top 5 positions, and larger for other positions. 

Google customises search results for users based on previous searches and clicked links. In other words, each individual's interests and search habits have an effect on search results he is shown. 

Google+ data such as your age, interests, gender, social connections and location all contribute to personalising your search results. For example, if you're logged into your Google account, you might be served search results that have previously been shared within your social network.
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