From my understanding of the rules it seems as though having initiative is actually a disadvantage. Initiative does two things when the players have pilots with tied skill- one is that the player with initiative activates his ships first, and the other is that the player with initiative fires first during the combat phase.
While moving your ship first can be an advantage, such as when trying to provoke a collision to cost the other player actions, activating after you see the opponent move allows you be better informed when choosing actions. More to the point, ships with higher pilot skill- which cost more squad points and therefore should be superior- activate later, so it follows that the game's default assumption is that activating after the opponent is an advantage. So in this regard having initiative makes your pilot act as though their pilot skill is lower than the opponent's.
During the combat phase the player with initiative will fire first. Firing first is usually a big advantage, as it allows you to potentially remove opposing ships before they can act. Initiative only matters in cases of equal pilot skill though, and by the simultaneous actions section on pg. 16 of the rulebook all pilots of equal skill get an opportunity to fire before being removed from play due to damage. It is possible that firing first would be an advantage in this case, as any ships that have yet to fire must abide by critical damage effects that may have been imposed upon them, but this requires that the killing blow both involved a critical hit, and that that critical hit affects attacking in some way (8 of 33 cards potentially could, with 4 of those being situational). Firing second give you more information to make your choice on, as you'll know if a ship might survive or not. This advantage is also somewhat situational, as it requires a set up where you might choose a different action knowing that a ship is about to explode anyway- firing a torpedo or blowing a target lock on a normal attack in range 1 instead of saving it for instance- but it is certainly an advantage. With this little bit of give and take having initiative in the combat phase a wash at best.
It's possible that having initiative is no supposed to be an advantage, so the rule just exists to give direction for tied pilot skill resolution, but there are a few things that suggest that it is supposed to be an advantage. First, generally in gaming winning or having the initiative is a good thing, and I imagine most gamers will find it somewhat contradictory that it's actually bad here (though as a side note the way it's used in x-wing matches a more literal definition of initiative). Second, in tournament rules it's the winner of the coin toss that has initiative, and again traditionally winning a coin toss provides an advantage, not a penalty. Finally, if players do not have the same squad value initiative goes to the player with fewer points. Squad value is a metric of how powerful your group is, and it really doesn't make sense for the player who brings a weaker (by value) squadron to the table to be additionally penalized.
Short version: Having initiative makes you activate and fire first in cases of tied pilot skill, but activating later is better and firing first is a wash at best since the other pilots in the same skill group all get to fire before dying. So initiative is a bad thing, which is odd.