The issue appears to be rather complex, so I do not aim at providing an exhaustive answer.
At a toy model level it is reasonable to model the eye as a "camera". Specifically, let us assume that a human eye "samples" at a maximum frequency of $\nu$, so that we may make use of the Nyquist-Shannon sampling theorem. Basically, given an instantaneous angular velocity of $\omega$, if the wheel has $n$ spacings, then the "highest frequency" component is $n\omega\over{2\pi}$ (i.e., in a full rotation, there are $n$ wheel bars passing at a given angle). Therefore, writing $\omega = {v\over r}$ with $v$ being the car speed and $r$ the wheel radius (here I am assuming pure rolling of the wheel), when
$$
v > {{\pi\nu r}\over{n} }
$$
we may assume that some kind of aliasing took place, i.e., I guess you would be unable to reconstruct correctly the wheel motion.
So assuming that a typical wheel has 10 bars and a radius of about 0.3 meters and your eye samples at ~30 Hz (typical frame rate of most first person shooter videogames, so it may be used as an upper limit since there one has complete illusion of movement), a rule of thumb calculation yields about 30 meters/second as a reasonable threshold speed for aliasing phenomena.
It's hard to make the wheels spin at high speeds because you're in a higher gear, so the torque at the wheels is less. So I assume you are only asking about wheel spin in first gear i.e. it's quite easy to spin the wheels when pulling away in first gear but much harder if e.g. you're travelling at 10 mph in first gear.
The reason is that if you're stationary and drop the clutch the angular momentum of the engine contributes to the torque. That is, the torque at the wheels is the torque from the engine plus the torque from angular momentum stored in the flywheel, crankshaft etc. This happens because the engine is spinning faster that it would if the clutch were engaged, so engaging the clutch slows the engine speed. The extra torque is given by:
$$ \tau = I\frac{d\omega}{dt} $$
where $I$ is the moment of inertia of the spinning bits of the engine and $\omega$ is the engine speed, so $d\omega/dt$ is the rate of change of engine speed. If you drop the clutch the engine speed changes rapidly so $d\omega/dt$ is large and the extra torque is large. If you ease the clutch out $d\omega/dt$ is small so the extra torque is small and the wheels won't spin.
When you're driving at (e.g.) a steady 10 mph the engine speed matches the wheel speed, so if you now suddenly stamp on the accelerator it's only the torque from the engine that's available to spin the wheels. You don't get the contribution from $d\omega/dt$.
To see this try driving at 5 mph, then disengage the clutch, rev the engine and drop the clutch. As the clutch bites the wheels will spin just as they do when the car is stationary.
It's worth noting that a powerful car can spin the wheels in first gear even without playing with the clutch. In fact an old sports car I had many years ago would spin the wheels in second gear in the dry and in third gear if the road was wet!
Best Answer
Because friction is your method of steering! (- and of braking and accelerating.) As @MasonWheeler comments:
Turning / steering
Friction is what makes you turn left at a corner: you turn the wheels which directs the friction the correct way. In fact, by turning your wheels you turn the direction of friction so that it has a sideways component. Friction then pushes your wheels gradually sideways and this results in the whole car turning.
Without friction you are unable to do this steering. No matter how you turn your wheels, no force will appear to push you sideways and cause a turn. Without friction the car is drifting randomly according to how the surface tilts, regardless of what you do and how the wheels are turned.
Braking and accelerating
Accelerating and braking (negative acceleration) requires something to push forward from or something to hold on to. That something is the road. And friction is the push and the pull. No friction means no pull or push, and braking and accelerating becomes impossible.
So, friction is very, very important in any kind of controlled motion of vehicles that are in touch with the ground. Even when ice skating, you'd have no chance if the ice was 100% smooth.
It should now be easy to grasp that it's a problem to go from static friction (no slipping of the tires) to kinetic friction (the tires slip and skid), simply because kinetic friction is lower than maximum static friction.
If you brake e.g., it is better to have static friction, because it can reach higher values than kinetic friction and thus it can stop you more effectively.