Mostly density and scale...

some compute tasks like serving a site are relatively trivial, think driving a car down a block

producing AI slop is very intensive, think disassembling the car, moving each part down the block and reassembling it all back together

A server farm might serve hundreds or thousands of web applications from a few dozen physical machines.

A data center uses hundreds or thousands of physical machines to provide highly intensive compute services to a few dozen applications.

The wasteful part (in my opinion) is that these few dozen applications are producing content that is not tied to reality in any meaningful way. Where they lack information, they don't seek it; they invent something that sounds good. Until AI is capable of accurately identifying and citing its sources, everything it says is gossip.

Another waste: We have applications converting a few pieces of data into lengthy reports. We have applications summarizing lengthy reports into a few pieces of data. Both applications are turning electricity into heat; neither one is needed, and both are regularly caught inserting their own gossip into the process.

The terms are kind of fungible. They don't have rigid technical definitions.

But a server farm could be doing anything. Serving up websites, hosting cloud storage, rendering movies, providing financial exchange services, scientific analysis, genuinely anything you might want a bunch of computers to do.

The main difference is scale.
You could turn a spare bedroom into a small server farm.
Data Centers are massive complexes, usually dedicated to extremely intense compute tasks. The most intensive computing right now being AI stuff.

I like your answer the best.

I’d add that server farms sprung up to cluster network services around high speed network cables; they allow individuals to locate their services somewhere that others can access them easily.

Data centres traditionally were run by single organizations to centrally store their data — where a server farms was about providing services in the most accessible way, data centres were about efficiently storing really large data sets in a single place.

AI compute centres are really both at the same time; they store massive amounts of data AND provide an interface to that data in the same place, because the compute services need near instant access to the data sets to function properly. So you end up with heavy compute clusters and data clusters running side by side, as close to the highest speed and bandwidth network cables as possible.