There have been many questions lately that focus on improving the quality of the image taken, where the answers circle around using a second-curtain flash to reduce blur and freeze motion.
Why not use second curtain-flash all the time? Are there down sides to second-curtain flash?
Second curtain sync doesn't do anything more to freeze motion than first curtain. The reason to use second curtain sync is that the ambient light trails you get with a long exposure appear behind the subject (as the sequence is ambient first then flash) instead of in front, which looks a bit weird.
In the top image (first curtain sync) the headlight trail appears in front of the train, in the bottom image (second curtain sync) the headlight trails burn in before the flash fires, so they appear behind the train.
Can't think of a general disadvantage of second curtain sync (I use it all the time), except that not all cameras offer true second curtain sync. Canon cameras for example only send a signal on the first curtain. Second curtain sync is then achieved by sending the shutter speed to the Canon compatible flash which knows to wait the specified time before firing.
One reason not to use second curtain sync might be that it's easier to frame a moving subject if you concentrate on where the subject is when you press the shutter, rather than where it will be when the shutter closes.
There are actually three possibilities here:
The last is also goes by a couple of other names: trailing curtain or rear curtain. Nikon, for example, calls it "rear curtain + slow sync".
Normally, the flash is so bright that shutter speed has a negligible effect on the exposure — the exposure time is the length of the flash pulse, because the amount of light let in during the rest of the time the shutter is open is insignificant.
Slow-speed sync modes combine a flash with a long enough shutter speed to also be recorded. This gives, in effect, a double exposure: a quick sharp exposure matching the flash pulse, and a long exposure matching the shutter time.
The difference between first curtain and trailing curtain sync is simply when the flash is triggered in relation to the long shutter speed: at the beginning or end, obviously. (It's interesting to read up a bit and understand exactly how a focal plane shutter works, but isn't necessary to the basic concept of using these modes.)
The disadvantage of using a slow-speed sync mode in general is that you introduce the same difficulties of any long-exposure photography, where both camera movement and subject motion are significant factors.
First-curtain sync has a distinct problem: since the motion-freezing flash pulse comes first and the long exposure after, if the subject moves, you get a ghostly image moving away from the sharp image. If you take a picture of a moving car, for example, the blurred line of the headlights will move in front of the camera. This looks weird to most people.
Rear-curtain sync avoids this, by capturing the trail first and then the frozen subject. This is also good for, for example, people moving by firelight — the "ghost" appears to track their previous actions, which better matches our perception and memory of the flow of time.
But there is a downside as well: you have to anticipate the action by the chosen shutter time. If you've picked a two-second shutter speed, the flash exposure will come two seconds after you press the shutter button. This makes it hard to capture an exact moment intentionally, and you need better luck, anticipation skill, and planning (and even that might not be enough in all situations).
A little bit esoteric, but second-curtain sync may not actually allow enough time for a complete flash discharge with some external (non-system) flash setups. This would occur mostly with battery pack-and-head systems with small flash tubes when using a lot of power (a Lumedyne setup with multiple boosters, for instance, or microphotography lighting) -- the resulting long flash duration usually means you'd need to set your camera to a shutter speed one or more stops slower than the X-sync even if you aren't looking for an ambient ghosting effect. That will work for front-curtain sync, but rear-curtain sync will not allow enough time for the extended flash. You could keep pumping up the flash power all day, but all you'd get is an overheated flash head.
There are a couple of more downsides for second-curtain sync not yet mentioned in other responses.
With TTL, flash power needed for correct exposure is determined by measuring reflection from a preflash before the shutter opens. Using rear-curtain flash, the distance between subject and the flash may have significantly changed during exposure and the pre-calculated value is more prone to be wrong.
Similarly, focus (especially auto-focus) is usually based on where things were at the start of exposure, not where they are expected to be by the end - you have more chances of missing focus for a moving subject using rear-curtain flash unless you have planned for it.
EDIT: Also, with most cameras viewfinder is black during exposure. Handheld, you will have harder time trying to keep desired composition and avoiding tilting the camera until flash fires.
