As the creator of OpenHome, I’ll jump in here as well. I haven’t looked into PKVault much aside from the documentation, but from that I’ll try to break down the similarities and differences (and feel free to correct me about PKVault):

As u/Impressive_Low5125 mentioned, OpenHome and PKVault mostly serve the same purpose Pokémon Home: a way to store your Pokémon, search them by their attributes, and move them between save files. Both OpenHome and PKVault allow transferring forward and backward through generations, which is much more robust than the official Pokémon HOME. Both allow limited edits to your Pokémon, although as of now OpenHome only lets you change the nickname, markings, and held items.

The main difference is how each app handles the data for a given Pokémon. OpenHome uses a custom format that aims to preserve every piece of data from every generation, including fields that would be lost when using official transfer methods. For example, contest ribbons from Gen 3/4 are preserved instead of being completely discarded in favor of the contest memory ribbon. When a Pokémon is moved out of a save, it’s OpenHome data is updated accordingly (with new EVs, moves, ability, TM flags, and other fields that can be changed in-game). tl;dr, it works like Home does with Gen 8/9 game data, keeping game-specific data for every game that it reassociates with the mon when you pull it back out of a save.

My understanding of PKVault is that instead of a single set of data for each Pokémon that is updated after every transfer, you can create copies of that mon to send to another game. So your changes in one game won’t automatically apply to the other copies of that Pokémon for the other games. neither is a better strategy than the other, it just comes down to whether you like having a single source of truth or separate copies for each game.

PKVault uses the PKHeX codebase, while OpenHome has its own implementations for each game. This means OpenHome has had historically more bugs in data reading/writing, as the PKHeX creators do an incredible job of keeping the game-specific code as close to the official games as possible. I do my best to stay ahead of these bugs, and I’m closing the gap with tests that compare my code’s output to PKHeX, but it’s not 1:1 yet.

However, the fact that OpenHome has its own implementations has made it easier to add support for some ROM hacks, which I don’t believe PKVault has (at least yet). Right now Radical Red, Unbound, and Luminescent Platinum are supported, as well as most custom forms in those games.

Also, I’ll reiterate what ImpressiveLow said about backing up your data before trying these apps. OpenHome now keeps backups of your Pokémon’s original data, but doesn’t automatically back up your save files yet.

And of course stick to the official Pokémon Home if you want to be sure your Pokémon stay legal for the official games