Newman's Skyfall score is more or less a contemporary sci-fi TV show score . . . mostly some unmelodic, low-key notes and chords with occasional bouts of a motif or percussion riff. His father was more traditional, and his score for such films as How the West Was Won is what I think of when I think of proper movie music. Modern audiences tend to find it over-the-top, but I find the minimalist modern scores mostly yawners. A few composers -- Hans Zimmer and Trent Reznor, for instance -- can pull of the minimalist thing, but most just create sonic wallpaper.
David Arnold is a blend of the two. When he uses a strong melody, such as Bond's arrival in the Bahamas or his escape from the Bolivian hotel, his stuff is good. He can write a decent theme, like his simple but effective "Sherlock." But his action music often sounds discordant and random, rather like a Michael Kamen score. It's not something to listen to outside the context of the movie itself.
What Barry understood, perhaps from his jazz background, was that even if the notes sound off-key, they had to somehow work together, not as random tones, but as a stand in for melody. Take his "007," which, like some jazz, is almost anti-melodic in the ways the notes come together, yet somehow, they give Bond a kind of urban, modern quality, the way jazz by Dave Brubeck, for instance, does.
Where Barry is strongest, though, is with action music. He uses simple but strong chords and lots of rhythm. Take his piece during the attack on Fort Knox, for instance. At the same time, he could write terrific, "pop" and jazz standards. Though "You Only Live Twice" sounds like a pastiche of Doris Day's "Once I had a Secret Love," just as "Goldfinger" sounds a lot like "Moon River," he still provided enough originality for the songs to take on a splendid life of their own.