The new book official 5e book, Mordenkainen Presents: Monsters of the Multiverse, contains a lot of the same content as the two previous books Mordenkainen's Tome of Foes and Volo's Guide to Monsters.
Currently (Feb 22nd 2022), Monsters of the Multiverse is only available by buying the complete Rules Expansion Gift Set.
Some of the content from Monsters of the Multiverse has been changed from VGTM and MTOF. For example, many races provide different benefits, and some of the monster stat blocks have been edited. No errata have so far been released to retroactively update VGTM and MTOF so they can match Monsters of the Multiverse.
When content in Monsters of the Multiverse doesn't match the versions of that content in older books, which rules take precedence?
Best Answer
Older versions of the races revised in Monsters of the Multiverse are obsolete. But ask your DM.
Your DM has the final say on which sourcebooks are available for selecting character options, and what rules from any sourcebooks are in use or take precedence. According to Wizards of the Coast, the versions of the races printed in Multiverse are the versions that "fit into the current state of the game":
D&D Beyond, now owned by Wizards of the Coast, gives a clear statement about the old versions of the races published in Multiverse, on their page about Legacy content:
And we see on the contents page for Volo's Guide and Mordenkainen's Tome this disclaimer:
So the position taken by Wizards of the Coast is that the older versions of these races are obsolete. From a "which ones are the most official" perspective, the races found in Monsters of the Multiverse take precedence. But from a "which rules take precedence at my table" perspective, it will always be the DM. This also tells me that it seems unlikely that these older books will be given errata to match the new publication.
As a final note, I haven't addressed monster stat blocks here because they are primarily DM-facing, and it's always been entirely up to the DM which monsters to use, so Multiverse hasn't really changed much for us, other than to provide some revised statblocks.
At the time of the current revision, the Adventurer's League Player's Guide (v11.0) had not yet been updated to account for the release of Multiverse.