Indeed Herbert himself has been said to have jokingly formed the "We’re Too Big to Sue George Lucas Society" when he recognized elements from his own works and many other writers in the film.

And let's be clear, Star Wars does not have the same story as Dune. It took elements and themes and paralleled them but not the direct plot. Dune itself took inspiration from The Sabre of Paradise.

If you wanted to be fairly uncharitable you could argue that A New Hope is basically Akiro Kurosawa's The Hidden Fortress set on Dune.

Spice Up Your Life

In the first draft of Star Wars, the drug ‘spice’ was very much a central theme of script.

Lucas's first version of Princess Leia was not fleeing from the Empire with the stolen plans for the Death Star but rather, holding a cargo of the drug 'melange' spice...the later script noted Han Solo smuggled of spice...

And let's not forget that young Luke Skywalker once had the belief that his "father didn’t fight in the Clone Wars, he was a navigator on a spice freighter.”

Here's a comparison of some direct lifts and inspirations

Dune's influence on A New Hope

• Princess Leia’s name is arguably inspired by the Princess Alia.
• Star Wars features a dry desert planet called Tatooine. Remind me what kind of planet Arrakis was in Dune? "I don't like sand"...
• The Jawa Sandcrawler was was possibly inspired by the mining vehicles Arakin’s used. We're iffy on this connection.
• The Skywalker family were moisture farmers in a similar vein as the 'dew collectors’.
• The 'Jedi Mind Trick' is very similar to how the sisterhood of the Bene Gesserit can use The Voice to influence the actions of others.
• In ANH, Luke Skywalker practices his lightsaber training against an automated training remote. This seems a direct lift from the part where Princess Alia works on her sword skills against an automated training dummy.
• Spice mining... Luke's father was a 'was a navigator on a spice freighter.'

The Empire Strikes Back was influenced by Frank Herbert

• In Empire Strikes Back, the Millennium Falcon just manages to escape from the jaws of giant, space slug that was living in a space asteroid. In Dune, there are giant sandworms a giant one of which causes a bit of similar havoc when one attacks the Duke's vessel. 
• In ESB, the villain turns out to be hero's father. In Dune, the villain out to be hero's grandfather
• Alia can connect her mind to her brother Paul Atredies physically. Kinda like what happens when Luke calls out to Leia after he's fought Darth Vader.

How Return of the Jedi was inspired by Dune

• Return of the Jedi’s Jabba the Hutt looks like a giant slug with a fat face and arms. This character may have been inspired by The God Emperor of Dune, Leto Atreides the Second who funnily enough was a 15-foot long slug. That novel was published in 1981, two years before the release of ROTJ. We do appreciate that Jabba was originally conceived in ANH as a human.
• Indeed, a lot of the whole Jabba The Hutt Palace scenes appear to have been inspired by the Children of Dune sequel.
• The sand desert where the Sarlaac Pit resides is called the Dune Sea

Mortal Engines author Philip Reeve shared his views on Dune influencing Star Wars when he was discussing the definition of 'space opera':

"Star Wars was, of course, a love-letter to the genre, full of motifs which come straight out of pulp fiction and the rockets hips’n’ray guns serials of 1930s cinema.

It also borrows heavily from Frank Herbert’s novel Dune, which I read when I was about twelve, mad about Star Wars and looking for something similar.

I was far too young to get Herbert’s mix of Orientalist fantasy, desert ecology and flaky 1960s mysticism but, as with Star Wars, that combination of swords and starships appealed to me. "

Let's not forget Lawrence of Arabia having an influence on Star Wars

Both Dune and George Lucas were inspired by the story of Lawrence of Arabia which was a spiritual journey of sorts that took place in a desert. Lucas was most definitely inspired by the David Lean movie and use shots from it for both the original and prequels films.

We say this to show that though the book came before the film, they both share common inspirations. Indeed, 19 other movies went into the creation of Star Wars.

We should acknowledge that a lot of Shakespeare's plays influenced Frank Herbert as he developed Dune - so while many of the family dynamics in Dune and the 'saga' of it all, can be compared and contrasted with Star Wars, one could perhaps argue the driving force of Shakespeare ripples underneath the surface quite strongly in Star Wars regardless of Dune's influence.

Indeed, to dismiss Shakespeare's work as an influence on George Lucas would be a mistake. Though Lucas may not have been directly taking paragraphs out of Shakespeare (unlike which he did for Tolkien!) his admiration of The Forbidden Plant exposed him to what was basically The Tempest set in space. Given Lucas's love of films by Akiro Kurosawa, he most likely saw 'Ran', which was a remake of King Lear!

Shakespeare actually has a lot to answer for - it was his line from Othello that inspired the name of the Mortal Engines novel!