The relays will switch up to 2A, whereas the Darlinton Array that supplies the outputs can only sink up to 500mA. The one you use generally depends upon what device you’re switching.
The Darlington Array outputs can also only sink current, they do not function as a power supply (there’s no 24v supply available on the Pi anyway) which means your wiring from the source, through your sink, and into the Automation HAT would all be live at the supply voltage if you were switching it with an output.
This is usually fine, but means the majority of your circuit is still prone to shorts when it’s off.
Supplying a circuit via a relay has the benefit of cutting power to the whole circuit path on the output side of the relay when it’s off, making it safer.
In most uses, however, your main consideration will be how much current you’re dealing with, or whether you need to source/sink interchangeably.
It’s possible to pair two relays, for example, to drive a motor in forwards and reverse directions. It’s not possible to do this with the Darlington driver without the use of additional circuitry.