In 1e, initiative determines who "strikes first". I played a 3.0 or 3.5 game with a friend, and it was all about weapon speed, and guys even got to strike first because they were using their fists. I'm all, WTF??? Something didn't sound right to me about that system.
Weapon reach can be even more important than weapon speed when it comes to determining who hits first. This is seen in the "charge to attack" rule and the "grappling" rules.
One page 66 in the DMG: "Initiative is NOT checked at the end of charge movement. The opponent with the longer weapon/reach attacks first. Charging creatures gain +2 on their 'to hit' dice if they survive any non-charging or charging opponent attacks which occur first [although their AC is one factor worse and they lose DEX bonuses]...."
Weapon length also has an effect in a grappling scenario (pg 73):
"If the opponent of a grappling, pummeling or overbearing attack has a weapon, the opponent will always strike first unless the attacker has surprise. Any weapon hit does NO damage, but it does indicate that the attacker trying to grapple, pummel, or overbear has been fended or driven off, and the attack is unsuccessful. The weapon-wielder then has the opportunity to strike at the weaponless one 'for real', if he or she so chooses. "
Weapon speed only comes into play for tie during initiative:
"When opponents in melee have tied for initiative, blows occur simultaneously, except when both opponents are using weapons. Each weapon has a speed factor, and in the case of otherwise simultaneous blows, the opponent with the weapon which has the lower speed factor will strike first."
Interesting extra factor thrown in here:
"When weapon speed factor is the determinant of which opponent strikes first in a melee round, there is a change that one opponent will be entitled to multiple attacks. " In sum, a speed factor advantage of 1/2 the other weapon, or total value of 5 better than the other weapon = two attacks for the speedier weapon, and a speed factor advantage of 10 provides 2 attacks plus 1 extra simultaneous attack.
Speed factors of common weapons:
Axe, hand -- 4
Club -- 4
Dagger -- 2
Hammer -- 4
Mace, footman's -- 7
Scimitar -- 4
Spear -- 6-8
Sword, bastard -- 6
Sword, long -- 5
Sword, short -- 3
Sword, two-handed -- 10