Very interesting to watch these two scenes side by side. In my view, there are a few differences between them in the way the events unfold. Some minor, some major. I like the extra bit of suspense in the Coens' version after LeBoeuf fires his gun.
However, there are choices that the Coens and composer Carter Burwell made that to me is truly baffling. The entire scene is scored with variations of the hymn
Leaning on the Everlasting Arms which throughout the movie is Mattie's theme. They chose this hymn partly to underline her religious conviction and Scott wrote about it
here in his True Grit analysis, pointing out the significance of this hymn and its connection to Mattie's journey.
The Coens' version is told from Mattie's point of view and like Jeff said, the original is focused on Rooster Cogburn. No matter which point of view one chooses, this particular scene should belong to Cogburn. His life is on the line and the audience should experience the events through his eye (pun intended) to get the maximum effect.
In the original version, composer Elmer Bernstein divides the scene up nicely in segments based on what the filmmakers want us to feel at each point in the story. After the “Fill your hands” line and the start of the shootout, we have no idea how this will end. The music doesn't take sides nor reveal the outcome of the fight. That makes total sense. Why would we want any spoilers as to what will happen? The music reflects that ambiguity to keep the suspense going. A few moments later there's a shift. A turning point, both literally with the horse and with Cogburn succeeding in shooting two of the Pepper Brothers and I'm guessing a third one off screen. The music shifts with the action and turns heroic, underscoring that Cogburn has the upper hand. Then the music shifts again as we see the wounded Ned Pepper take aim and fire at Cogburn.
Like it or not, this is the point of most film music. To tell the audience how to interpret actions unfolding, to help them along so there won't be any misunderstandings, to tell people how to feel. When we hear that rousing theme there's no doubt that Cogburn's got this one in the bag. Therefore when the music turns sinister as Ned Pepper takes aim, we feel it much stronger. The music has successfully manipulated us into thinking that everything is okay and that the danger has disappeared.
Now, compare that to the Coens' version with Carter Burwell's score. It starts out with a hint of the hymn – a mere outline of the melody played on the violins during the lead-up to the fight. Then, as the shootout starts, we get a couple of introductory chords and then the hymn again orchestrated with big heroic brass. Here we already know the outcome. Cogburn's the hero. We can hear it in the music. There's no ambiguity to what the outcome will be. No suspense.
Note how in the original the cut to Mattie and LeBoeuf up on the hill (Shoot them... Too far...) occurs during the first part of the shootout, before Cogburn hits anyone, before the turn in the fight. In the remake, the cut happens after Cogburn hits the first guy and after we hear the heroic music thus diminishing the suspense. We already know the outcome.
But the most confusing thing is the use of the hymn - the hymn that belongs to Mattie - as score for this particular scene. We hear the theme based on that same hymn a third time as Cogburn is stuck under his horse and Pepper pulls out his gun. If the purpose of having music at all in a film is to help guide the audience through actions and more importantly emotions, then this choice is truly mind-boggling. Why do we get Mattie's theme when the scene belongs to Cogburn? Whose POV are we suppose to experience? What are we supposed to feel?
Now, most of this happens on a subconscious level. We're not supposed to notice the music. We don't want to be aware of it. We've all heard film music that is too manipulative, too melodramatic, too much. It ruins the drama and takes us out of the movie. Here, in this scene of the remake, the music choice only serves to confuse the audience. And even if most people won't notice it, it's there, and it seeps into our subconscious whether we want it or not. The result is the feeling of uncertainty (and not in a good, deliberate way) and the feeling of not enough guidance from a strong hand that should lead us through the story. The result is storytelling that is unclear because we're not sure which character to identify with.