The Princeton offense and the triangle offense are first cousins.
Born from the principles of offensive spacing, floor balance and reading/reacting to what the defense is doing, both offenses are designed to create good looks every time down the floor. By taking advantage of the mismatches that present themselves due to ball movement and the subsequent defensive rotations, there's a reason both offenses have been as successful as they have been.
All that said, even though there are similarities between the offensive schemes, the way the Lakers will run the Princeton offense in 2012-13 will differ from the ways past incarnations of the team ran the triangle. Here are a couple of ways how:
How to Use Pick-and-Rolls
While the triangle used pick-and-rolls as secondary actions within the offense, this year's Lakers will use the pick-and-roll as a way to initiate their primary offensive sets:
Here you see what will be a typical offensive initiation by the Lakers. Nash starts by bringing the ball up the left side of the floor but rather than passing the ball to the wing, he calls for a pick-and-roll to be initiated with Dwight Howard. After the screen is set, Nash sees that Howard is covered so he swings the ball to Pau Gasol who then lobs the ball to Howard for a dunk.
This differs from the triangle in that even when the pick-and-roll was used it was always after a guard-to-guard entry where the primary action was executed that would evolve into a pick-and-roll:
In this triangle set you see the Lakers start their possession with Odom bringing the ball up and then passing the ball to Kobe to create the first guard-to-guard pass. Odom then runs what's called a UCLA cut off the high post and then cuts to the opposite low block. After receiving the ball Kobe then runs a pick-and-roll with Gasol but you can already see the difference between how Kobe is attacking versus how Nash attacked in the clip above.
Kobe is obviously probing the defense and setting up an action on the weak side rather than looking for anything primary on the ball side. Kobe swings the ball to Fisher who then hits Odom with a quick post-entry pass. After Odom catches the ball, he finishes with a nice little jump hook.
The differences between how today's Lakers use the pick-and-roll in the Princeton offense and how the team used it under the triangle may seem subtle but that's not really the case. Today's Lakers will likely pound teams with the pick-and-roll to start their offense many times throughout the game whereas the Phil Jackson teams would mostly use it after sets had already started and not as the primary action.
Strong-Side vs. Weak-Side Handoffs
Another of the main actions the current Lakers will use as an offensive action will be a strong-side handoff series:
In this set you see Nash bring the ball up on the right side of the floor where Kobe and Dwight Howard are situated. Nash begins the set with a high-post entry and then proceeds to go set a screen for Kobe. Kobe comes off Nash's pick and then darts towards Howard where he receives a handoff. After getting the ball from Howard, Kobe knocks down a jumper against limited pressure.
This differs from the triangle in a few ways. First, again, you see the ball entered straight into the high post rather than with the guard-to-guard entry. Next, the ball-side handoff is not a typical action that the triangle incorporates. That type of hand-off action is typically reserved as an action on the weak side:
Again, this difference is subtle but meaningful. Rather than keeping the ball on the strong side and using screen actions to set up a hand-off sequence that puts the defense in a difficult position to choose who to guard, the triangle used the threat of an open weak side where space could be taken advantage of to create the shot.
In the clip above Shannon Brown settled for the long jumper, but if that were Kobe, the odds are that he would have taken the handoff from Bynum and looked to turn the corner into the open space that was available along the right lane.
Because the philosophies and principles of the triangle and Princeton are so similar, the likelihood is that there will be similar sets run in both offenses. However, the the 2012-13 Lakers will surely shape the Princeton to their personnel, and that will create the types of differences you see above.
To the naked eye they may seem like only small wrinkles but to a defense game-planning on how to slow the team down, these changes are substantial enough that old game plans simply won't apply.