Does the API have the same 1,000-row limit?

No. The Search Analytics API returns up to 25,000 rows per request and supports paging, though anonymised queries are still withheld.

Why do my clicks total less than the chart?

Because the rows you can see exclude both the rows beyond 1,000 and every anonymised query. The chart counts all of it.

Why your Search Console export stops at 1,000 rows

You exported your queries, got exactly 1,000 rows, and the numbers do not add up to what the chart said.

Short answer

The Search Console interface caps a query export at 1,000 rows, and separately hides queries searched by very few people to protect privacy. Your totals will not match the chart because of both. The gap is real data you cannot see in that file, not an error.

Two different things are missing

The 1,000-row cap is a limit on the file. Anonymised queries are a limit on the data itself: Google withholds rare queries entirely, and no export or API call returns them. The first can be worked around. The second cannot.

How to get more than 1,000 rows

Filter and export repeatedly: by page, by country, by device, by date range, or by query substring. Each filtered export gets its own 1,000 rows. Or use the Search Console API, which returns up to 25,000 rows per request and is free. Or connect the bulk data export to BigQuery, which delivers everything daily with no cap.

Why the missing tail matters less than it looks

On this dataset, the top five URLs of 227 carried 75.2% of all clicks and the top twenty carried 95.6%. The anonymised tail is long, and it is thin. Decisions are made in the rows you can see.

What to do before you export

Set the date range deliberately, because the file is a slice of a period rather than a current state. Then split brand from non-brand before you read a single average: on this site brand queries ran at 31.55% CTR against 1.94% for everything else.

Questions

Does the API have the same 1,000-row limit?: No. The Search Analytics API returns up to 25,000 rows per request and supports paging, though anonymised queries are still withheld.
Why do my clicks total less than the chart?: Because the rows you can see exclude both the rows beyond 1,000 and every anonymised query. The chart counts all of it.

Measured, not asserted

Free tool for this: Search Console Analyzer. No account, nothing uploaded.

Where this goes deeper

Every number on this page comes from one complete dataset: one product taken from zero to 250K+ users on search alone, with nothing spent on advertising. The full argument is appendix-a-the-complete-dataset of the book. Five chapters are free to read.