Text Diff lines up an old version and a new version and marks every line that was added, removed, or changed. On a changed line it also highlights the exact words that differ, so a one-character edit is easy to spot.
The comparison runs in the page with the browser's own JavaScript. Neither block of text is uploaded, and closing the tab leaves nothing behind.
No. The diff is computed by JavaScript in the open tab. There is no upload and no stored copy.
No. The diff is line based, so a moved block shows as a delete in one place and an add in another.
A standard unified diff with a configurable amount of context. It applies with patch or git apply.
Thousands of lines are fine on a normal laptop. Tens of thousands use more memory because the whole comparison is held at once.