Halo 2's weapons, compared to Halo 1's and perhaps compared to other shooters, work or combine in such a way that the average time of any given encounter is greatly reduced. This makes any latency you are experiencing a greater percentage of the total encounter time than it would be if it took longer to get a kill, and places a lagged player with dual SMGs going up against a player also with dual SMGs is at a greater disadvantage than if both players had single SMGs.
Other, more diligent Halo fans have already compiled hard data on this, showing the time required to kill with various weapons and combinations under various conditions, including melee kills and whether or not a weapon gives special damage for head shots. In some cases, the time required for a kill is less than 1.5 seconds. If your ping is above average, say, 250ms, then you are behind your opposition by more than 15% of the total time required to kill. If you are similarly armed, similarly accurate, and engage each other simultaneously, you will lose. You need to be more accurate, better armed, and/or begin firing more than 15% earlier than your opponent in order to give yourself the same chance at victory than a player with a lower ping has.
The sword, shotgun, sniper rifle, and rocket launcher are, if used properly, all essentially one-hit-kill weapons. Going up against a similarly armed player (as in a games of shotties or swords or rockets) means, that if players are equally accurate, the one who shoots first wins. This is nearly always the player with a decent ping compared to anyone over 250ms, as if you fire earlier (i.e., further away) you're more likely to miss and thus lose in any case.
Various combinations create a similar effect; the use of the plasma pistol either with the BR, the human pistol, or the SMG, greatly reduce the time required for a kill and thus increase the advantage given to the player who fires first.
However, some elements of those combinations, as well as grenades, are likely to cause you some frustration because of the way Halo 2's network model works. When you toss a grenade or fire an overcharged plasma pistol, you'll see the set of arm-animations almost immediately. You'll see your arm move as if you're tossing the grenade, but the grenade itself won't actually appear, move, or explode until the server recognizes that you've thrown it-- by which time you and your target have both moved. The same goes for the plasma pistol bolt; its range and homing effectiveness is apparently reduced from the first game, so to be sure to get a hit, you have to be closer. But since you're counting on that landing to lower your opponent's shield and get the quick kill with another weapon, you have to make sure it's a hit. It's very easy to charge in with the plasma pistol and then go for the easy kill, only to be shown mere fractions of a second later that your plasma bolt never landed, meaning you're toe to toe with another player with full shields.
Encounters in Halo 2 end differently and more quickly than those in Halo 1. Given the player's relatively small amount of health in Halo 2, both compared to the original game and compared to the maximum level of Halo 2's shield, combined with the faster recharge of that shield, means that confrontations are driven to a conclusion faster than before. Long standoffs and evasions are less likely; encounters are more likely to be decisive, ending either in a clear victory for one player and the quick restoration of that player's health (given time and cover), or in mutual annihilation of two or more players.
As a player with a high ping, you're going to wish you got at least a fractional score for Assists; especially if you're avoiding the difficult-to-use one-hit-kill weapons in favor of dual wielding. You're going to do a lot of damage but not get as many kills. The trick is to try to do your damage at the end of encounters rather than at the beginning.