Why do airports have to close in bad weather?
Wind

The problem with wind is that it can be from any direction, not necessarily the direction of the runway. When the wind is coming from the side instead of straight ahead, it is called a crosswind. This means that landing aircraft have to point into the wind to remain lined up with the runway. The slower they are flying, the further to the side they must point the nose. Airliners are designed to land in these conditions, but it requires more skill to do it well. The control inputs required to compensate for a crosswind affect aircraft on takeoff as well, as then must work harder to stay lined up with the runway and stay level through lifting off. High winds tend to come along with gusts, which makes it even harder to control an aircraft. Aircraft during flight are flying faster, so changes in wind are less problematic. Once they are flying slower, the wind is much larger compared to the aircraft's speed, and control is much more critical during landing. Aircraft have crosswind limits, and even below those limits, aircraft may need to go around from a landing attempt, which introduces delays.

Another issue is wind shear, which is when the direction and/or speed of the wind quickly changes. A change of 40 knots at higher speeds is turbulence. During landing or takeoff, it can cause the aircraft to stall and crash.

Visibility

Aircraft can certainly use systems like ILS that allow them to take off or land in low visibility. However, there are many limitations with these systems. The crew must be trained to use them, the aircraft must be equipped to use them, and the airport must be equipped with them. However, when the pilots are relying on instruments, they no longer are able to see hazards like usual. Because of this, separation requirements are inreased in these conditions, which means an airport can't handle as much traffic.

Water

It's true that aircraft wheels are not directly powered. However, traction is still very important during braking, since aircraft can hydroplane just like a car can. This means aircraft need more distance to stop, and can slide off the runway if they lose traction. Frozen water in the form of ice or snow makes a plane even more difficult to steer or stop.



Combine all of the above (a typical storm that would cause delays at an airport), and you can see why airports delay flights or close when conditions get worse.