Frequency of AC is 50-60 Hz, it is quite fast for our eyes. In addition, light emission is due to either Joule heat or luminescence. In case of Joule heat it is black body radiation, which depends on temperature. And the filament cools down much slower compared to changes in the current. Luminescent is faster and one can see the blinking in a video if the framerate is fast enough.

Update. To sum up good comments to my answer.

A very well formulated answer by @Chris and some real world numbers by @NightLightFighter cover the physics of the question very well. Here are some different aspects of it.

Time factor.

The frequency of AC current is 50-60 Hz hence the frequency of the electric power is double that. As Chris says, the filament of incandescent bulb is heated by the electric current and radiates light as any heated body does.

Intensity factor.

When the power is at minimum, the filament cools down mainly by radiation (thermal conductivity of the filament is not too good and convection is not very effective especially because of the protective glass cover). The characteristic time of this cooling is significantly slower compared to the period of the electric power, so the change of the filament temperature is not too high and causes only small change in the radiation intensity. According to @NightLightFighter it is only

~5-10% of its total brightness.

Fluorescent tube.

Again thanks to @NightLightFighter

A fluorescent tube has a variation of about 40% of its total brightness.

Biological factor.

we don't see with our eyes, we see with our brains.

Human vision is a complex system and not everything is understood about it yet. However, capabilities of our vision are well studied and the actual answer to the question is that they are not good enough to see the light flicker of the light bulb (in most cases). In this study published in Nature, Scientific Reports, 2013, it was found that humans can detect some changes in the displayed picture on a monitor which are spatially uniform with threshold of the modulation of about 63 Hz. And for images with some sharp spatial edges even up to 500 Hz!

I assume it is safe to think of the incandescent bulb as of a ‘spatially uniform’ image. Speculations are that the changes in the light intensity are much smaller compared to the average value and they happen at a rate of ~ 100-120 Hz, which is higher than our capabilities, according to the cited research.

In case of the fluorescent tube, the changes in the light intensity are significantly bigger and one can assume that this case is closer to images with ‘spatial edge’ from the research. Therefore, the detection threshold could be higher. This may be the reason why some people can have headaches with such lighting.