In contrast to the available build tasks in my personal VSTS instance:
I have another (seems like older) set of tasks available on my client's TFS instance (e.g. a ".Net Core (Preview)" task):
What does the TFS admin have to do in order to update these? Where should I point him to in order to get to know why the set of available build tasks is different?
In order to update these, you have to upgrade to the latest TFS 2017 (Update3) or later version (TFS 2018...). Update the task independently is not supported.
Generally all the latest new features will be available first in VSTS, then integrated to on-premise TFS later. In the other hand there are some differences between the features available in the two products. Some TFS features are not supported in VSTS at all—for example, VSTS does not support integration with SharePoint or Project Server.
Some preview features added in earlier version may not be stable enough, so TFS team will test the feature and release the stable one in next versions.
Refer to About VSTS and Team Foundation Server for the differences between the two products.