The language I would use to differentiate between the two is that one is a Commercial Solution, the other is an Implemented System (you could argue my terminology here as well).
ArcGIS Online is a cloud hosted GIS Content management solution, that can be used different organisations.
That is -
- Its in the cloud
- It publishes content which consists of map services (layers), apps, analysis tools, downloadable datasets etc.
- Has security mechanisms such that only content that you want available to selected users will be available.
So a practical example could be a mining company hosting and publishing data as Feature Services, that are available to be consumed as layers into ArcMap or web mapping solutions, specific to their organisation. No one outside the organisation can see these content items.
Another example could be a government department hosting downloadable utility data available for download by the wider public community.
With your specific example, when you 'Add data from ArcGIS Online' you would also normally see datasets/maps/content that are being published by your specific organisation that you are currently working in. Or - public content such as what you are showing above.
Living Atlas of the World - is what I would describe as a specific custom Data Catalogue and Spatial data publication system. Its purpose for an end user is to enable easy searching of data,maps and apps (ie: Spatial content! as above with ArcGIS online). And its all Public. So anyone can use it....Its also designed for system administrators to catalogue and publish data/maps/apps in a nice neat way. In your example, there is no 'Add Living Atlas of the world' button inside of ArcMap because its custom.
One of the differences between the two is that:
- ArcGIS Online is a commercial solution provided by ESRI, (just like how ArcMap is a solution. )
- Living Atlas is an example of that solution (and other technology ) being put into practice (ironically by ESRI).
A comparative example could be said that if ESRI sold you ArcMap you would just get the software. And then if ESRI was publishing ArcMap MXD's that you could download, which had all layers and data with them and searching capability and they called it the 'ArcMap Living Atlas' - then that would be a comparative example.
To your specific questions:
Do they contain the same data - Yes - But not 100% like for like....... Some data may be on Living atlas, but not in ArcGIS online, and vice versa. Depends on how its being published.
Is one a subset of the other? Sort of. Living Atlas is (probably) using ArcGIS Online solution to host a lot of its content. You could say that the Living Atlas system uses the technology solution 'ArcGIS Online'. But it would also use other technology solutions together to make the system.
If they are different, why can't I access the Living Atlas from the ArcMap Add Data dropdown? - ArcGIS Online, being a commercial solution, is consistent technology... That is, if I buy it, you buy, its the same underlying technology which means that ESRI are able to build a 'Add Data from ArcGIS Online' button into other solutions knowing that ArcGIS Online everywhere is the same technology.
But the Living Atlas is essentially a custom solution, one example of many thousands from around the world (data.gov.au is another comparative public example). Each of these are a website that has been deployed by an individual organisation, which means each site is different in its technology implementation........ If that organisation wanted a feature inside of ArcMap (or any software for that matter) then they would need to build it, for that specific site.
Unfortunately ESRI are quite fluid and dynamic with their solutions and so it can be a bit confusing sometimes. Terminology is also inconsistent across industry (some people call it a portal, some an SDI, some a catalogue, some a content management system etc etc) so your confusion is not unwarranted!