This fantastic post over at IMDB explains the differences in a lot of detail (given its length, I've chosen to just link to it rather than copy and paste it in its entirety).
A summary of some differences (note, the link above is far more comprehensive and descriptive). This is intended for all people who have seen the US version and are wondering how the UK version differs:
Around 22 minutes in (after the Horowitz scene), there is an extra
scene where Jake explains in the car to Avi and Zach his history with
Around 1 hour in (after the scene in which the guy chasing Jake accidentally shoots himself), instead of the restaurant shoot-out scene, Jack goes to Paramount City to talk to Avi and Zach. The restaurant shoot-out scene follows this.
During the Driving Range scene, there are a few extra comments inserted.
Around the 1 hour, 16 minute mark (after Driving Range scene and Jake giving money to charity), instead of Mache scene where we start to hear his thoughts, the actual structure of the film is changed. Due to the complicated nature of what happens next, I will quote from the above article:
We see the Machiavelli quote, then we have the scene where Jake is
standing in Macha's bedroom, admitting his inferiority to Macha (the
climax of the US version). This's plays out just like the US version
does. Instead of the US epilogue (where Jake and Avi play chess and
talk about Jake's enlightenment), after the "Fear me" scene it cuts
right into the "crucifixion" of Lord John's man and the scene where we
can start to hear Macha's thoughts. Originally, this marks the natural
transition of Mr. Gold's thoughts from Jake (where he is now
powerless) to Macha who has just been humiliated by Jake and deep in
fear of Mr Gold (So it now makes more sense why we now can hear
Macha's thoughts). Instead of the torture montage, Macha brags of
Jake's groveling the night before and that he "knows who's boss"
(conveniently forgetting to mention the "fear me" scene). Once Paul
tells him Jake was behind everything, Macha has a very brief flash of
the "fear me" scenes played back in his head (yelling instead of
whispering the words) instead of the "they don't fear me" monologue
the US version has. The UK version omits the "Sam Gold will kill me"
internal monologue and replaces it with one of the whispering/crying
"fear me" scenes.
In the UK version, instead of letting the Sorter shooting spree scene
play out like seen in the US version, we get an over the phone
retelling of the event by the surviving flunky Lou to Macha. He says
he got tired of waiting at the car, went inside, saw Paul getting
carried away with the little girl, and "from out of nowhere, Sorter
shot him" (the dialog we hear in the US version is not in the UK
version thanks to the voice over of Lou). Macha is in disbelief. "Then
he turned the gun on the 2 bodyguards, both gone in a blink of an
eye." The rest of the scene plays out the same in both versions. At
the end of Sorter's killing spree when he is wiping his glasses, Lou
is the one who kills Sorter by shooting him in the back of the head.
Lou says he was the only one left. Macha asks "where's the girl?"
before the scene cuts to the exterior of Macha's casino.
The next scene (which is Jake’s final meeting with Macha) is different as well.
When Macha tells the clerk to bring Jake up, this is where the US
epilogue is (where they talk about Jake’s new enlightenment). This is
played out with added content and much more detail, to give the
viewer a better understanding of what is (meant) to be happening.