Differences:

-The PS2 version loads much faster (on emulator). The PS2 console seems slightly faster but that will have to be looked into.

-Flying heights work differently. In the Gamecube version, you may switch freely between 3 heights. In the PS2 version, the middle height is the default, and you raise or lower your characters by holding down a button and the height resets when the button is released.

-The PS2 has a longer chemical X recharge rate. The recharge on the Gamecube version is roughly 2 seconds. The recharge rate on the PS2 version is roughly 5 seconds

-The PS2 camera seems worse. It seems to be zoomed in closer and gets caught on objects frequently. I have not played the PS2 version for more than a few minutes so this may just be something you get used to and learn to control better.

-The PS2 Flying seems tighter. I'm not positive about this one but turning on the ps2 version while boosting seems tighter. This may be a preference thing.

-The PS2 Melee range is decreased? I'm not sure about this one but it seems like the range for melee attacks is drastically smaller on the PS2. It may also be another factor that makes it seem much harder to hit melee attacks.

-The PS2 version has fewer missions. The PS2 version does not have the mission where you chase a truck with a transmitter on it in Level 1. The ps2 version also does not have the mission after the Mojo Jojo escort mission in level 3, where you are required to take out a very large number of enemies. Both of these missions are long, the second one being one of the longest missions in the game.

-The PS2 does not specify in any missions which number of which types of the enemy must be defeated. It seems like any enemy works, whereas the Gamecube version specifies certain amounts of certain enemies in a few missions. The PS2's total number is one lower than the combined totals of the numbers in the Gamecube version. The Gamecube version also seems to have a preferred order on the mission with the pods which is done out of order can cause enemies to respawn and the numbers go back up if you don't follow the arrow. I am unsure about the other missions like this. This could potentially make these missions faster on PS2 if it does not have the respawning issue, or if these missions require fewer enemies defeated on PS2, or you don't have to look for the specific enemies and can use any enemy to meet your goal. Although with the chemical X recharge rate being faster on Gamecube, Gamecube combat may go more quickly. This is something that would have to be looked further into.

-The Anti-pickle weapons are cosmetically different between versions. I'm not sure about the function.

-PS2 Emulator crash. The PS2 version crashed at the same point for me in level 1 right before fighting Ace twice in a row, so I could not continue. I talk more about this in a guide post I made here: https://www.speedrun.com/the_powerpuff_girls_relish_rampage/guide/b4we4

-The PS2 version has minor weird sound glitches. (Don't seem important). (Emulator, unsure about consoles).

-The PS2 sometimes takes a second to load mission dialogue. Sometimes you can continue to move in this time. It is hard to tell but this could save a few seconds over the run because of extra movement, or it could lose time because of the longer loads.

-In the PS2 version, if you finish a mission before the starting dialogue ends, you can continue to move. In the Gamecube version, your characters stop until the dialogue is over. This saves a few seconds for the Ps2 version.

-Gamecube audio and visuals seem better on emulator versions.

-The PS2 version pauses music when you pause the game and starts the track over at the beginning when you unpause.

Unless glitches or skips are found that are version exclusive, the Ps2 version seems to be significantly faster, however, the Gamecube version has more content to run and is a better experience. Since the boards are split I recommend running the Gamecube version but it's up to personal choice.

Note: I am unsure if the "Pickle Edition" of the game makes any difference to the gameplay or mission count on either version. The game does not seem to have any indicators of which version you are playing other than the box art, so it is hard to know without physical copies. It does not seem like the "Pickle Edition" impacts gameplay at all on the Gamecube, I'm not sure if the ps2 version is updated with it or not, this would have to be looked into.