What ArchiveTeam mainly does is provide hand-made scripts to aggressively archive specific websites that are about to die, with a prioritization for things the community deems most endangered and most important. They provide a bot you can run to grab these scripts automatically and run them on your own hardware, to join the volunteer effort.

This is in contrast to the Wayback Machine's builtin crawler, which is just a broad spectrum internet crawler without any specific rules, prioritizations, or supplementary link lists.

For example, one ArchiveTeam project had the goal to save as many obscure Wikis as possible, using the MediaWiki export feature rather than just grabbing page contents directly. This came in handy for thousands of wikis that were affected by Miraheze's disk failure and happened to have backups created by this project. Thanks to the domain-specific technique, the backups were high-fidelity enough that many users could immediately restart their wiki on another provider as if nothing happened.

They also try to "graze the rate limit" when a website announces a shutdown date and there isn't enough time to capture everything. They actively monitor for error responses and adjust the archiving rate accordingly, to get as much as possible as fast as possible, hopefully without crashing the backend or inadvertently archiving a bunch of useless error messages.

I just made a root comment with my experience seeing their process at work, but yeah it really cannot be overstated how efficient and effective their archiving process is

Their MediaWiki tool was also invaluable in helping us fork the Path of Exile wiki from Fandom.