Oh Oh! I think you opened the "box of pandora" with this question ^^.
Ok, so... Batocera is a fork of Recalbox.
One (or more, i don't know exactly) team-members/developers of former Recalbox got in a twist/fight with another one (or more) about the way Recalbox should be evolved/developed and decided to to his (their) own thing and so created batocera-linux. Or he decided to do his own thing parallel to Recalbox and then got into trouble, i don't know
Someone was accused of stealing code or to have stated something under his name, which is not his right (or something like that).
That is as far, as i was understanding (please correct me, if i'm wrong, I only heard the whole thing in fragments, see here for example: https://batocera-linux.xorhub.com/forum/d/815-shame-on-recalbox).
For me as an outsider everything was very confusing and i still do not know, what happened exactly.
As for Raspberry Pi, the differences between Recalbox and Batocera are not earth-shattering in my opinion.
Recalbox has netplay (which i only tested a view times and i'm not really using it, because it has too much lag and therefore is not really fun) and web-manager (with some usefull features, but i'm not missing it in Batocera, and for RPI2 and RPI3B (without +) in my experience, it causes too much cpu-load in the background, so that for example you can not use Run-Ahead Feature in some games, which also demands cpu-power. But you can deactivate the web-manager).
And Recalbox has a more modern Theme with more features (also more themes in general).
In most of other aspects, for RPI, Recalbox and Batocera are almost the same and are both good. Best is, you make your own experience and decide for yourself.
But where Batocera is definitely better, is the PC-Version (x86_64).
In the newest version you have Saturn, PS2 and even 3DS support (with newest beta).
Batocera in general also gets more/faster updates than Recalbox.
For example Batocera supported RPI3B+ already since 01.05.2018 (so about a half year faster, than Recalbox.. oh, and Kodi 17 has also been part of Batocera for over a year or so).
I hope this post does not get deleted by one of the moderators here.
If so, i apologize for writing this all.
Please do not understand this post as an accusation or criticism.