Ok, so it is important to understand that the underlying design principles for all product development at Atlassian is determined by the Atlassian Design Guidelines V3 (or ADG3), which can be found at https://atlassian.design
On that page, you will also find a very limited subset of React components maintained by the Atlassian Design System team (DS). Unfortunately, the cool kids of the Design System Team are either completely understaffed by Atlassian or are far to cool to care about anything apart from that limited subset.
As a result of either their coolness or lack of support from Atlassian engineering leadership, there is also an extended set of React components that can be found at https://atlaskit.atlassian.com. These components are often special purpose and created by Atlassian product teams to solve specific problems for the product. You will find banners saying that these components are either experimental or for internal use only. They are still published, you can still use them, but those warning basically means nobody at Atlassian gives a crap about them and that they are poorly maintained. Ye be warned, there will be dragons if you use those components (even though some of them are core product components like the editor or media components).
Now the very cool kids of the Atlassian Design Team are either to cool for Server/Data Center or have been told by leadership not to care about those products because the only thing they do is bring in a shit load of , so another team within Atlassian is responsible for AUI, which is the framework that is used for Server/Data Center products. Surprisingly, this team is extremely active to a point that you wonder why they are not called the Design System Team. Unfortunately, they are bound to the quirks of Server/Data Center and are still using jQuery and AMD-style loading of dependencies. So not very cool, but it works. They are also slowly migrating Server/Data Center to ADG3, while still carrying the burden of the limitations imposed by outdated architectures.
Anyway, long story (and somewhat unveiled rant) short: if you’re only going to develop for Atlassian Cloud, and can either make due with a very limited subset or are willing to pull out your hairs screaming at your screen (and the void), you should go with DS components in combination with AtlasKit.
If you’re only going to develop for Server/Data Center, I would go with AUI.
If you’re developing for both… good luck to you sir!
At some point Atlassian told vendors to use AtlasKit on both Cloud and Server (because future), only to abandon most of AtlasKit and leaving vendors out in the cold after most of them followed up on that recommendation.
To be honest, at this point I consider both DS and AtlasKit as dead code