The human ATC system is very good at handling exceptions, including various kinds of emergencies, pilot errors, and reasons to prioritize one plane over another. The human controllers also typically have a good understanding of the details of airspace, regulations, policies, aviation customs, and the capabilities of various kinds of aircraft.
So for example, a plane can have a "missed approach and go around" when trying to land. I was once on a plane that did that because of extremely high winds. In that case, the plane that was supposed to be on the ground is suddenly climbing again and is going to need to turn in order to repeat the approach.
A plane can have a medical emergency onboard, so it needs to land at an unexpected airport, possibly faster than a normal landing, or starting from a somewhat atypical position. I was once on a plane that did that because someone onboard had a seizure.
A plane can have damage or equipment failures that the pilots find it hard to assess directly, so it needs to fly around for a while to give the pilots time to "run checklists" or perform various tests, or sometimes to let a ground-based observer report on something about the plane (!), or just to develop their intuition about how functional the plane is. It might then need to land at an unintended airport or return unexpectedly to its takeoff airport. I was once on a plane that did that because of bird strikes during departure, where the plane was damaged but the pilots were unsure how seriously.
A pilot might misunderstand something or disobey regulations, and that pilot or another pilot might be told to take some unexpected evasive action to avoid a collision. (This is one area that has been productively automated in some cases via TCAS, where the aircraft themselves can sometimes figure out what an appropriate maneuver would be before a controller tells them one.) I haven't personally experienced that.
There might be another kind of emergency where a runway is closed and a large number of planes need to be diverted (like right after this collision where DCA was abruptly closed).
There might be negotiations with an uncooperative or mentally ill pilot (like the tragic story of Richard Russell in 2018, but also a number of incidents that had happier endings).
Pilots might also negotiate more cooperatively with ATC related to diversions and priority in situations like bad weather, where the airport has less capacity than originally expected and the pilots need to determine whether they will divert to a different airport. In this case the air traffic controllers may talk to different pilots about their fuel levels and other factors that make them better and worse candidates for changing flight plans. En-route (ARTCC/ACC) controllers will also negotiate with pilots about changing altitude to reduce or avoid turbulence.
There are occasionally cases where a pilot is incapacitated and someone with less training and experience needs to be advised remotely on how to land a plane. (This is mostly very small planes but ATC will still ultimately deal with these emergencies.) In that case other planes also need to be kept away from the incident aircraft and maybe diverted elsewhere.
Specifically for takeoff and landing, there are often multiple planes using the same runway (for takeoff, landing, or both) in relatively quick succession, or possibly using runways that cross each other. In this case, a controller needs to keep an eye on how quickly pilots have (or haven't) complied with specific clearances, e.g. to cross a runway on the ground, because the clearances may need to be revoked or modified if they aren't used quickly enough (because of the presence of other aircraft that have also received clearances that will soon start to conflict with the older clearances). This also includes checking whether planes that have landed have vacated the runway expeditiously (since if they haven't done so, for whatever reason, other planes may soon need to be told to go around).
There are also cases where military or law enforcement authorities may ask or demand to modify normal ATC procedures or clearances because of some special operation or problem. The simplest case is that they might ask to prioritize a government aircraft over civilian flights for some reason, or ask certain other operations to stop e.g. during a takeoff or landing of Air Force One. (I just watched this a few days ago with an Air Force One departure from Las Vegas, where other departures and landings were temporarily but briefly suspended. So that had to be planned and communicated to various pilots, some of whom then had follow-up questions about what they were or weren't allowed to do.)
Pilots are also considered to have ultimate responsibility for the safety of their flights and passengers, and they can also refuse some ATC instructions, or deviate from some normal procedures, in emergencies. So for example, a controller might believe it's safe to land in certain weather conditions and might give a pilot a certain clearance, but the pilot might not feel up to completing the landing and might then refuse to do so. The controller will have to understand the pilot's intentions as best as possible, and deal with the consequences of those intentions (e.g., once again, keeping other planes out of the way, or trying to find a new routing that the pilot will be willing to accept).
ATC is also responsible for passing some kinds of information to and from other parties, like in case of an emergency landing communicating with emergency responders so that they understand the nature of the emergency and whatever facts will help them respond more effectively. And they have to tell other ATC facilities about problems and situations that will affect them, like in-flight emergencies, closed airspace, closed runways, closed airport, etc.
Many of these things can and should be more automated than they are, but humans in these jobs are doing enormous amounts of reasoning, improvisation, and even social negotiation.
(I'm not a pilot or air traffic controller, just a former frequent flyer who liked listening to ATC communications and sometimes listens to liveatc.net when friends' flights are arriving or departing, or watches video recaps of various aviation incidents.)
Edit: Another case that I thought of: during an emergency landing, a pilot might be given either a shorter (more direct) or longer (more indirect) route than usual, in response to the pilot's assessment of which would be safer. The pilot could also be given a longer route than usual in order to have time to "work checklists" in preparation for the landing, or in order to burn off fuel so that the plane will weigh less (and be less likely to cause a huge fire) upon landing.
If some navigation equipment is broken, the ATC facility could help with navigation or with diagnosing the problem (by describing visual landmarks, or by estimating the plane's current speed and heading based on ATC radar).