Vanilla is the official Minecraft server software from Mojang. It supports no plugins, has limited performance options, and receives official updates the fastest.
Use vanilla if you want a pure, unmodified experience with no plugins.
Spigot is a modified version of vanilla that supports plugins via the Bukkit API. It offers some performance improvements over vanilla and has a huge plugin ecosystem.
Most plugins you find on the internet will work with Spigot. However, Spigot's development has slowed significantly.
Paper is a fork of Spigot with massive performance improvements. It's fully compatible with Spigot plugins, runs async chunk loading, and has hundreds of configurable performance settings.
Paper is the recommended choice for almost every server. It handles more players, runs at higher TPS, and gives you more control.
Use Paper. It's faster than Spigot, fully compatible with Spigot plugins, and actively maintained. The only reason to use vanilla is if you want zero modifications to the game.
In the Lyrebird control panel, you can switch between Paper, Spigot, Forge, and Fabric in the Version tab with one click.