Google results are not consistent:

Different geographies (even in different cities within the United States) often give different results.

Different data centers give different results (and you can hit multiple data centers from a single location at different times).

Results are personalized for logged-in users based on their search histories.

No rank checker can monitor and report all of these inconsistencies (at least, not without scraping Google hundreds of times from all over the world with every possible setting).

The Google API rarely matches up to what anyone sees in the search results:

It appears to match up only on very heavily trafficked, consistent search results— anything mid-tail or long tail is invariably inaccurate.

Obsessing over rankings (rather than traffic) can result in poor strategic decisions:

When sites obsess over rankings for particular keywords, they often spend much more time and energy on a few particular key phrases that produce far less value than if they had spent those resources on the site overall.

Long tail traffic is very often 70% to 80% of the demand, and it is much easier to rank in the long tail and get valuable traffic from the long tail than to concentrate on the few rankings at the top of the demand curve.