"toughest" goes in cycles. Whenever there is bad weather, weird things start to happen. We had 22 flights diverted from DFW to SAT one night. Ops has to find room to park them all! This gets even crazier during special events, like Final Four, when lots of private jets are coming in.
In colder cities, they have to contend with ice and snow. It's a lot of work to keep an airport operational. Or, there are problems when pilots misjudge a turn and get stuck in the mud. Or, an aircraft's landing gear collapses on the runway, putting the runway out of service. I would say all of this is more likely to happen in the fall/winter.
As far as things in the terminal are concerned, holidays and summer are more hectic, because the people travelling are less likely to be regular travelers, so they get more stressed out.