Can someone explain why initiative is so important when 2 ships with the same ps will both get to fire at each other before damage is assigned? What is the advantage of shooting first?

The Simultaneous Attack Rule simply says that in this situation a ship has the opportunity to attack before it is destroyed. It does not say to hold off on assigning damage. One advantage in shooting first is that you might land a crit on the ship.

It's also important when you're moving. Move first and you can block your opponent and deny them action/position. Move second and you can line up your shot.

So that crit will take effect while the opponent takes their shot?

So that crit will take effect while the opponent takes their shot?

Yes, the crit is applied when it is dealt. Some may not come into play till next turn such as Console Fire and Damaged Sensor Array, since they happen before attacks. But Weapon Malfunction will, so any return fire suffers the effect.

Thanks all, makes more sense now

Initiative matters in 3 places:

Turn 0: The placement of ships and asteroids

Movement: If you have initiative, you move first

Combat: If you have initiative, you shoot first.

Shooting first in Combat lets you:

1. Strip focus tokens from them by forcing them to defend first.
2. Potentially land a Crit, debuffing them.
3. Potentially kill one of their Allies that would buff them, like Howlrunner.

The effects in the other two zones, however, are much more potent.

Another thing that happens frequently is TIE Phantoms equipped with an Adv Cloaking Device. With initiative, it would get to fire and then immediately cloak before the opponent fires back, thus making it 4 agility. Initiative is only used to resolve timing issues where both players have something happening at the same time. Sometimes it's good to have it, and sometimes it isn't. When the PS is equal, the player with initiative always moves first and always shoots first. There's no choice on that.