By comparison, here's the Dvorak, and here's the Williams. The similarities between the two are, to put it kindly, extremely superficial. Both start with low strings intoning a note, and then the note a half-step above it, and then the motif is repeated a few times. But Dvorak repeats it loudly and uses all the lower strings and goes at a quick tempo, building quickly and bringing in the rest of the orchestra before getting to his main theme. He also stays quite clearly in the same time signature.
Williams, however, starts off with similar notes...but slower, and much softer, and lower -- I'm not even sure if he uses the cellos at all. It might be just the double basses at first. And then his insistent rhythm starts with those punching chords at off moments, so you're not even sure what the time signature of the piece is. Williams's sound is insistent and mysterious and somehow both mechanical and not -- pretty much the opposite of what Dvorak does. And yet, "Williams ripped off Dvorak!" is one of those zombie nonsense notions that always comes back, despite being complete nonsense to anyone who bothers to pay attention.
In cases like this, for years I've been recommending a wonderful essay by Leonard Bernstein called "The Infinite Variety of Music", which appears in the book of the same title. The essay is actually the script of one of the wonderful episodes he used to do for the educational teevee program Omnibus. In this particular episode, Bernstein described how composers are able to create an astonishing variety of musical works from just thirteen notes of the Western tuning system, by reducing things even further and showing how a number of great composers wrote amazing pieces, many of which are very familiar, by using as their main motif the exact same four-note melody. It's a worthy reminder that there's a lot more to music than just what the notes are, and I've always found that essay to be a good remedy against the over-used canard that this composer or that composer ripped someone else off.
Of course, the problem with recommending an essay like that is that it's in a book that isn't always readily available...but I've recently discovered that the audio of that very program is on YouTube, with the musical examples helpfully included so you can see what's going on as Bernstein speaks. I can't recommend this highly enough. It's certainly worth the 48 minutes to listen through. No, Bernstein doesn't specifically address Dvorak or Williams (in fact, this program was likely recorded while Williams was still a studio musician and Steven Spielberg was a kid), but it does suggest a good way of listening to music to evaluate such silly claims.
Here's the video:
Really, give it a listen. It'll make you better at listening to music!