I just read that a normal flash illuminates a scene within a 1/250th of a second. (A flash would keep the scene illuminated for a 1/250th of a second, right?

In general, that's wrong.

Flash duration is flash duration and sync speed is sync speed. Apples and oranges.

The 1/250th of a second is the sync speed of (many) cameras. That's basically the shortest shutter speed at which the shutter is entirely open during the exposure. For shorter shutter speeds, both curtains of the shutter move at the same time, which means that the exposure happens through a moving slit formed by both shutters.

The flash exposes the entire scene. But if there's only a slit of the shutter open during the flash exposure, only that slit of the image will be brightened by the flash - the result is two (or three) horizontal parts of the image with different exposures. (either only ambient light or ambient light + flash)

In summary: To get all the image exposed by the flash, the shutter has to be open entirely when the flash fires. The shortest shutter speed doing that is called the sync speed.

And it's the flash's sync speed, right?

The sync speed is a property of the camera, not the flash

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Flash duration on the other hand is the time the flash is on. A flash is not just on or off. The power (light) output of a flash is a curve. Flash Havoc has a great explanation in their article about the Phottix Odin Over Drive Sync timing:

A flash pulse from a studio strobe may look very fast to the human eye, but its actually often a relatively long process, consisting of a bright initial burst, followed by a long slow exponential decline, or long tail of light. (Think of a firework that goes off with a bright burst and then slowly dissipates). So flash duration curves will often appear as shown below, a short peak followed by a long tail of declining light.

This flash duration curve at full power nicely illustrates this flash curve and the two different ways to measure the duration of that curve.

The problem is that for many flashes, the flash duration depends on the power setting. If you change power, the flash duration varies.

Andy Gock tested a bunch of flashes (and strobes) in his blog post for their flash durations.

Let me quote the results for the Canon 580EX:

Power μs s 1 4000.0 1/250 2 1088.0 1/919 4 484.0 1/2066 8 266.0 1/3759 16 166.0 1/6024 32 105.6 1/9470 64 71.6 1/13966 128 50.4 1/19841

As you can see, the duration for full power (1) is actually 1/250th of a second. But for all other settings, it's shorter.

The durations become so short because the different power settings are achieved by turning the flash off prematurely. This cuts off the curve and thus results in shorter flash durations. Andy did a great job illustrating all of this in his blog post.

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And(then) why can't we use a 1/250th of shutter speed to freeze motion in a well lit place?

A well lit place will not lend itself to freezing motion. That's because the ambient light is strong. The stronger the ambient light is, the more influence it has on the overall exposure.

In order to freeze motion with a flash, the influence of the ambient light (which causes blurryness) is reduced as much as possible, by reducing the shutter speed as much as possible, which usually means to shoot at the sync speed and in a dark environment.

Even with a flash duration = sync speed, the flash still fires in that curve, which means during the exposure, at the time when the curve is at its peak, the flash output is very strong. This period of strong flash will have a bigger influence on the overall exposure and can make a difference in terms of freezing motion.