In a sentence, I'd say apsw is "the SQLite lover's SQLite library". The python stdlib one tries to provide a generic interface ("DBAPI") which is great if you want to be able to swap to a different database engine later, but it's bad for actually getting the most out of SQLite.
This library is great, and recently saved a project of mine. Figuring out what was going on with some of python's sqlite3 quirks was too much of a hassle, meanwhile this more or less just worked the way I expected it to. Because of this experience I started defaulting to it over sqlite3 for other projects as well.
(This documentation page explains the library's intent: https://rogerbinns.github.io/apsw/pysqlite.html)