There are multiple reasons why you would want to synchronise firing times, however the largest reason why would be because of the effect of having one arrow/musket ball being fired against a hundred arrows/musket balls being fired. Whilst arrows and later bullets are deadly and do kill, the main purpose of having archers and gunmen is so that you made the opposing army or unit question themselves - disrupting their formations, impeding an attack, poking at them to keep them uncomfortable, these were all very valid reasons to have archers. However, none of these results could happen as effectively if archers (and gunmen - I'll group both under archers for the sake of simplicity) loosed their arrows one by one.

Morale played a large role in deciding the winner of a battle - in most battles, people didn't start dying until one side broke and ran, giving the other side liberal opportunity to run the fleeing enemies down and kill them or keep them for ransom. If you had a hypothetical unit of a hundred men, and you were advancing on an enemy unit, a single person near you dying to a stray arrow would not be nearly as morally devastating as if a dozen fell. You wouldn't doubt yourself if only one fell, but if ten fell then you'd wonder if you really should be charging into the enemy. Remember that, for the most part, units were effective as, well, a unit - cohesive, disciplined, and trained soldiers working as one. There are many examples of how these units can overpower an untrained horde of enemies, such as the Battle of Watling Street between the Romans and a mass of native British people - the Romans were outnumbered close to twentyfold, and yet still managed to prove victorious due to the discipline of the Roman army, and their ability to work in units.

Another reason why archers would loose their arrows in unison would be to make sure that everyone loosed together (via peer pressure or group dynamics), and to ensure that no one would be hurt on the same side. This could go for within the archery unit itself, as you do need some space to draw your bow, however it also applied when you were engaging an enemy unit already locked in battle with your own forces. If you simply did what you wanted to do and loosed as you do, you could end up hitting your own soldiers - however, if you all loosed at the same time, with minimal effort you could avoid a large friendly casualty.

To a lesser extent, when you fire en masse at a target, say, infantry, they will most likely try and defend against it, whether by moving as fast as possible out of the way or to raise their shields. This provides an opening for your melee units to engage, as the enemy would be busy protecting themselves from the arrows. This also applies for muskets, however with muskets the tactic to simply fire at will was slightly more justified in areas (for example, as a last stand, or if you were a relatively small group), however firing in volleys was still preferable. Line infantry would fire, then crouch down to reload their guns as the second line would fire, then the same as the third line fired, and so on in a cycle to maintain the volley. Similar to how it worked with bows, your volley fire could then closely be lead by a charge with bayonets, as the enemy is stunned or shocked and trying to regather their forces.

As a historical reenactor, there are several factors I can share.

1. limited ammunition. Early cartridge boxes have blocks in them, in which a little hole is bored. In that hole goes the cartridge, and there you have limited ammo. 12 rounds at the low end, 40 rounds at the high end. With arrows, the number is limited by your supply. Maybe again, 12-20 shots would be average. Commanders are able to control the rates of fire and maximize effectiveness, keeping track of their men and how much ammo they have left, as well as planning how to best apply the effectiveness of the units under their command. Allowing a fire at will from the beginning would lead to men running out quickly.

2. Accuracy is an issue. Up until the american civil war, accuracy was not very good at range. Regular infantry often used smoothbore muskets, and accuracy was quite a skill. Rounds were not meant to be wasted on a missed target, but also a soldier in a battle doesn't have all the time in the world to aim. Also we take into account the experience of each soldier. Marksmen are not as common as untrained shooters, and giving 100 men weapons and saying go kill someone would not be as effective as controlling 100 weapons and orchestrating your effective power as a unit. A commander would be able to point in a general direction, say FIRE or LOOSE and be cognizant of an expected effectiveness of his troops. It helped leaders make their moves and plan in advance. For instance - I have two units approaching, from the left and from straight on. I can fire two at the larger unit, fire one at the smaller, and shift position into a better angle, for my next volley. With fire at will, there's no control like that.

3. Soldiers advanced in ranks. So we've got soldiers who are not necessarily experienced marksmen, with weapons that don't have pinpoint accuracy in average hands, how can we get them to cause the most damage? Have them all aim at the approaching mass of men, and pull the trigger at once. We see them fall, they don't necessarily have a place to duck or hide or dodge to, and we reload, waiting for their attack. We also know that they're sending a volley our way, so in our ranks, we have strength in numbers. We hope that the volley will hit someone else. You can't really single out a target at range, you just act as a large organism.

4. There was a time and a place for each kind of fire. There was fire by battalion, fire by company, fire by platoon, fire by files from the right, and fire at will. Each came with different pros and cons, balancing the damage power, by the rate of fire, and the usage of ammunition. If your orders are to keep constant fire on someone, just to keep them pinned down, or prevent them from advancing, firing by files from the right was cool. The 2 soldiers in ranks at the far right fire, the ones next to them fire, then the ones next to them fire, and so on and so forth, until a wave of musket fire shoots down the line of men. The default action after this approach was to commence firing at will. It's like ripping off a bandaid, followed by gentle rubbing by sand paper. Firing by volley is like a smack, while you wait for their smack. Firing by platoons or by ranks was like flicks, which occurred more often than the smacks.

In the end, they're all just different tactics for causing damage to the other side. The armies are like single organisms. How can one fight them best?

Now, with the advent of accurate guns, camo, guerrilla warfare, we don't have any of those previous limitations, we hide, shoot. kill. we still maneuver, and direct impact and force in carefully managed ways, but attacking in ranks and shooting in volleys, was proven to be disastrous in the American Civil War, with rifled guns that fired so straight, and were so deadly, that you were sure to hit and be hit, and the only protection was to have massive, massive quantities of soldiers, so that you just hoped by the luck of the draw, today wouldn't be your day.

There is misinformation in this thread in that in volley fire all men in a 'unit' would fire at once. Volley fire would prevent exactly the problem of everyone reloading at once. If it takes 2 minutes to reload a weapon, and the range of your weapon can be covered on foot or by horse in that time, as more heavily armed troops will stay at a distance until everyone has fired. If some people haven't fired, they can make it very bad for people to come into closer range

Thanks to u/bigluepanda and all the other responders in this thread. I love learning the answers to questions I never thought of.