In many cases the differences between a ring and a cage would have a huge effect on the outcome of the fight. For example Randys style is perfectly suited to the cage and he would be extremly less effective in the ring.
There are 6 main things that I think affect the Pride guys and the outcomes of their fights in the UFC.
1. - Clinching aginst the cage - In pride there is virtually no pushing against the ropes and clinching as its just to unstable. This style of pushing your opponent aginst the fence is widely utilised in the UFC as a way of landing strikes and setting up the takedown. Therefore the pride guys are completly inexperienced at this major aspect of fighting in the UFC. Example, Mirko getting destroyed in the clinch by Kongo.
2. - The amount of space in the octagon and the shape of the octagon - In Pride the ring is relativly small, and forgive me for stating the obvious but it is square. This means it is much easier to stay out of a strikers range in the UFC and it is much harder for a striker to corner there opponent as there are so many ways out. Its just alot easier to retreat in the UFC. Example Mirko not being able to set up his high kick because Kongo could just backpedal all the time.
3. - Cage grabbing / pushing off the cage / pounding people on the cage - Fighters can grab the fence alot less obviosly in the UFC and until recently have been able to get away with it alot of times. Also in the UFC fighters can push off the fence with there hands as well as there feat which greatly aides in reversing position etc. Example rampage jackson reversing position on Dan Henderson. This could not be done in Pride as fighters were simply moved to the middle of the ring, quite a difference. This also meant that Pride fighters didnt have to deal with being pounded against the cage and have no experience in that sitauation. Example Mirko vs Gonzaga.
4. - Pressure - Even though these guys have been fighting good opponents infront of loads of japanese fans, they are not used to the way the UFC does things. They have to worry about being booed and establishing there reputation amoung a whole new group of fans, as well as worrying about representing Pride as a superior organisation. All this added presure has to effect them a great deal.
5. - Ringrust - Alot of the Pride guys hadnt fought for ages and the UFC guys had fought recently and had come off easy wins. Example Forrest vs Shogun.
6. - Drug testing.