Minecraft Spigot is a server-side modloader that loads a kind of mod called plugins, which can only modify the server, IE “server-side”. This limits what Spigot can do, but any Vanilla client with the same version of Minecraft can connect to the server without needing to download the same mods or configurations, which is often the case for modloaders like Forge and Fabric.
Paper is a fork of Spigot, which means it’s “forked” off of the tree that is Spigot and makes changes to Spigot. These changes include an extended API for plugin developers and enhanced performance features for server owners. We generally recommend the use of Paper over Spigot, and even over Vanilla even if you aren’t playing with any plugins or mods.