Why doesn't the brightness of a bulb change with time?

Two reasons:

  • An incandescent bulb glows not (directly) because it has electricity going through it, but because it is hot. Even when the power going through the bulb decreases, it takes some time for the filament to cool down. Even once the bulb is turned off, it takes some time (a fraction of a second) for the light to fade.
  • What variation there is in the light is too fast for our eyes to see.

You can see the AC flicker in slow motion videos if the camera has a sufficient frame rate, for instance this one.

We already have great answers, but I'll add an anecdote to this:

I once played around with several light sources and a scientific camera. The overall experiment was not overly scientific, and I only had one example of each light source, but it may serve as a starting point.

I found that:

  • An incandescent bulb only has a flicker of ~5-10% of its total brightness. This is due to its thermal mass, which keeps the filament glowing.

  • A fluorescent tube has a variation of about 40% of its total brightness. Its plasma and fluorescent coating retain some brightness, until the next current wave hits.

  • A cheap LED with a rectified input current, but no capacitor to even things out, really drops down to zero brightness.

As an addition to chris' answer

the intensity does change, but it changes very quickly and not by that much

The change is enough to be measurable, and in some ways, it can be really handy.

In the days of the gramophone (the large black discs that were used to play music before the CD), some record players could be fine-tuned to turn at the correct speed, and could be switched to turn at 33 or 45 rpm. In order to finely adjust the speed, the base of the turntable had a pattern on it which would appear to stand still when the table turned at the right speed. Often there were 4 patterns, one for 33 and 45 rpm for 50Hz and 60Hz electricity nets each.

This youtube shoes how this works using the little light next to the turntable, but the patterns where originally made to be used with normal incandescent bulbs - and it worked.

This site gives a bit more explanation.