Ran by different people and have different goals, derpi and ponybooru want to make a community, twibooru wants to be an archive, and ponerpics looks like it's going for both.
in terms of rules
Derpibooru: Strictest rules for comments and posts, shat the bed hard enough with rule changes, and complete lack of transparency to get the alternative boorus to start running.
Ponybooru: Replacement to derpibooru by being an almost exact clone of it, to the point where it's ruleset has 80% of derpi's rule text for word while also being almost double the length, only difference is comments and posts are slightly less moderated, staff use a forum post as a moderation log for what they do, and lolicon is allowed but you have to use or make a custom filter to be able to see it.
Twibooru: Comments forced anonymous, you have to have an account to be able to see them, forums completely disabled, they allow all artists' content with no restrictions, they allow anything pony related as long as it's not illegal or real life animal gore, stricter on furry content compared to other boorus, they have an automated moderation log for image deletions, they list their finances, and have a warrant canary.
Ponerpics: No dnps for public content, paid content dnps only hide the content for a time instead of removing it from the site, no form of moderation log yet.
I'd recommend twi and poner