Here are the reasons for a discrepancy between what you see in Google SERP and what Ahrefs shows in Site Explorer:
Timing: Google SERPs can change pretty much any minute (or at least every few days). At Ahrefs, we strive for efficient use of our resources, so we currently do not update keywords faster than a few times per day. We also reserve the more frequent updates for the most popular keywords. We do update the longer tail keywords but at a less frequent interval, in terms of weeks.
Location: SERPs can be different based on location. We do have IP addresses all around the world, but we do not have them in every city. This can cause some discrepancy in the SERP.
Personalisation: It's also not a secret that Google serves different results to different people with the goal of helping people get better search results (Read more.) This can be based on factors like their search history and browsing patterns. For e.g, your neighbour may get a slightly different SERP for the same keyword due to the above factors.
Any of the above 3 reasons or a combination of these can cause a discrepancy between what you see in Google SERP and Ahrefs' Site Explorer.