Paper itself is a series of patches over the official vanilla server implementation. You can convert a vanilla server to a paper server just by replacing the vanilla server.jar file with a paper server.jar one, and converting back to a vanilla server can be done fairly easily (but a bit more difficult than dropping in)
It's much, much more performant, and able to handle a lot more players/chunks loaded/etc on the same setup compared to a vanilla server. Along with significant performance improvements out of the box, it also adds more configurations (vanilla has server.properties, but papermc adds paper.yml) which allow you to hand tune optimizations[1], from things like disabling player collisions, disabling block updates for certain laggy blocks, etc. These changes are configurable because they trade off expected vanilla behavior for better performance.
The real strength of paper though is that they do all this while also implementing the spigot API, which enables paper server owners to use a wide variety of server mods[2] that can allow you to add things like minigames, anticheat, etc while still being compatible with unmodified clients.
Most would consider it to be vanilla, when "vanilla" is used as a descriptor of the gameplay (i.e., as opposed to "modded"). There are some very minor differences, but almost anyone advertising a vanilla Minecraft multiplayer server is going to be using Paper or one of its predecessors/competitors. The released server.jar just really isn't optimized for more than one player.
Some things that you can tweak for performance reasons also affect gameplay. For example, you can tweak the radius at which items group together into stacks of entities. Enabling this can make certain farm builds not work, e.g. a piglin bartering farm might have the gold bar destined for one piglin merge with the bar destined for a neighbouring piglin, when in pure Vanilla settings that might not happen. Similarly, you can have XP orbs merge into single orbs with higher XP values. But this drastically changes how long it takes to use an efficient XP farm, where normally you would be waiting for minutes for your player to receive the XP one orb at time.
I much prefer Fabric server to Paper. Paper has a couple game-breaking differences in advanced redstone, and Fabric in general has more mod compatibility. There are options to disable the redstone differences, but it's a bit annoying.
Basically just enable the unsupported settings. My players were happy with the end result of each one I flipped on so presumably the behavior was as expected.
It's much, much more performant, and able to handle a lot more players/chunks loaded/etc on the same setup compared to a vanilla server. Along with significant performance improvements out of the box, it also adds more configurations (vanilla has server.properties, but papermc adds paper.yml) which allow you to hand tune optimizations[1], from things like disabling player collisions, disabling block updates for certain laggy blocks, etc. These changes are configurable because they trade off expected vanilla behavior for better performance.
The real strength of paper though is that they do all this while also implementing the spigot API, which enables paper server owners to use a wide variety of server mods[2] that can allow you to add things like minigames, anticheat, etc while still being compatible with unmodified clients.
[1] https://www.spigotmc.org/threads/guide-server-optimization%E... (outdated but a bit more succinct) https://paper-chan.moe/paper-optimization/ (more technical)
[2] https://www.spigotmc.org/resources/categories/spigot.4/