Invite for havva sent!
I’d say the two main reasons why these leagues exist are: 1.: Having a ladder-esque system in place keeps things fresh and interesting for people who have been playing for a while - It gives older players and new players a chance to start on equal footing in a new environment with a fresh economy that doesn’t suffer from the inflation that you see in legacy leagues (and believe me, it’s nuts… standard/legacy especially, since loot doesn’t simply disappear, hardcore characters constantly die and add to that, etc. and people have been accumulating wealth for 9 months or so). They also add interesting challenges that are not (yet) present in the default leagues, which brings me to the second reason.
The Four-Month-Leagues usually differ from the normal/legacy leagues in at least one fundamental way and they add ‘stuff’ that you simply won’t find in the default leagues (yet) or remove items from the loot table that you can usually find in those. So apart from being interesting it also is a way for the developer to test new ideas or the impact of some items on the game without forcing those changes on everybody right away. So some of those changes will eventually find their way into the default leagues.
On the same note: Characters created in one of the special leagues will eventually end up in the default league anyway - Once the league ends characters and your whole stash gets moved to standard or hardcore, respectively - So in general i’d recommend anyone to simply start in one of those new leagues since there really isn’t a reason not to, unless you have a friend who outright refuses to create a fresh character there, of course.
So in short: new content, some minor changes in available loot, fresh economy.