The 1933 original which I watched for the first time today is generally (universally?) seen as the best version, and indeed, one of the best films of all time. This view is seriously rose-tinted. It's all about empathy.
In the original, Kong is a wild beast. He's animalistic. He's insensitive. He lacks empathy. While pursuing the fleeing Ann Darrow in both Skull Island and in New York, Kong eats (or at least gnaws at) a fairly off-putting number of people. He just picks them up, chews them for a moment then flings them to the ground. Essentially, Kong is a more accurate - if you can even use 'accurate' in describing a fantastical creature - representation of what a giant ape would be like. That could be fine handled differently, but it presents a few narrative problems. Kong's attraction to Darrow is no more profound than a magpie's attraction to a shiny coin. He is fascinated by her whiteness: all previous sacrifices have been black girls who are apparently only attractive enough to eat (the 1930s were less than racially aware). He wants no more than to hold her in his mighty fist, and never looks at Darrow beyond her whiteness.
Because of this, Darrow shows no emotion towards Kong other than fear. She doesn't interact with him at all - in fact, she says almost nothing for the remainder of the film once taken by Kong, instead she shrieks and shrieks. Well renowned film critic Richard Corliss from Time Magazine says Darrow came to sympathise with the beast. This is simply untrue: there is no evident emotional connection between Darrow and Kong. She says nothing (verbally or physically) to him, and even in the climactic scene atop the Empire State Building her anguish is not for Kong being slowly killed by the biplanes but for being stranded 381 metres above the streets below supported only by an out-of-control beast. Jay Antani, who mounts the most persuasive case against the remake, takes the view that the tragedy of Kong is his role as 'an outcast/schoolyard bully who falls in love with the local beauty–a woman who will never accept him'. How two-dimensional is that? Their entire dynamic can be explained in one scene, with no further development needed: Kong likes woman, woman resists. It doesn't change for the rest of the film. Dull!
There is an empathetic collapse here, and the problems with that are obvious. There is no emotional pull for the audience. In the remake we, channeled though Darrow, slowly come to understand Kong. As a character he is far, far more complex than in the original; the Kong we see in the original is the same Kong that the supporting characters see in the remake - it is through Darrow's interaction and gradually developing empathy for Kong that we come to understand him. This window is lacking in the original.
The second problem with having no connection between Kong, Darrow and the audience is that the action scenes have no emotional weight. The T-Rex fight may be animated with great skill, but without empathy for Kong we don't know whether we should be rooting for him or not. Sure, if the T-Rex wins Darrow will get eaten, but it's not clear whether Kong has dinner plans for Darrow or merely wants to keep her around to do...what, exactly? Manhandle her until she slowly starves? That doesn't sound a whole lot better than getting eaten by a dinosaur.
In contrast, the T-Rex fight in the remake is driven by character. Rather than just being a fight scene in the shallowest sense, it is a way (a really awesome way, I might add) of developing the Darrow/Kong relationship. Kong is enraged that Darrow has escaped, so sets of in pursuit and in doing so winds up having to defeat three T-Rexes while keeping her safe. At this point we are far more invested in both Darrow and Kong than in the original: we see Darrow trying to entertain Kong, we see her gradual understanding, and we understand the simple enjoyment Kong gets out of watching her perform. By making it clear that the audience is on the side of Darrow and Kong, the fight becomes far more engaging: Kong has to viciously fight these enormous creatures while keeping her safe. In comparison with the original, Kong has a far greater task.
Furthermore, once the fight is over, Kong does not simply snatch Darrow up and carry her off again, he gets upset. She has offended him. The contrast between his bravery and power in battle and his sensitivity afterwards is fascinating. Perhaps this was enabled by technology: Kong is as expressive as any human. Would this emotional detail have been possible with the crude methods available to the original? In that film a close up of Kong's face caused ripples of laughter among my coursemates - he looked like a gurning moron. The remade Kong has infinitely more depth. I would argue that this is a rare instance of technology actually being put to use for narrative purposes, instead of just making things looks better.
Lastly, a few other quick points. As neatly articulated by Jay Antani, 'in the original, Driscoll was the ship’s first mate; here, he’s Denham’s screenwriter and, hence, the Denham-Darrow-Driscoll triangulation is made more taut.' Yep. The biplane fight is shot far better than in the original. Kong is shot from further out, emphasising his impotence against the aerial threat. Denham is actually really good, even played by Jack Black. In the original he's stiff and uninteresting, in the remake he's wonderfully flawed. Jack Black is very unheroic and quite ridiculous by nature and it suits this new Denham perfectly. There is also the question of whether Darrow loves Kong. It's not obvious, and I'm not sure of the answer.
There. That was my attempt at doing some serious analysis, Film Crit Hulk style.