Because the King James bible is written in deliberately archaic but still fundamentally prosaic English, while Shakespeare is intentionally difficult. The Bard is showing off, all the time - not for the benefit of educated aristocrats, if anything it might have actually bothered a significant number of them - but because puns and classical allusions and wordplay were a huge part of the way he wrote. Some of its difficulty to modern ears is the fact that we lack a lot of the cultural general knowledge his audiences would have had which would make these coherent at first glance - and any joke you have to have explained to you or look up is no longer funny - but people were criticizing him for this in the 1700s, so some of it is definitely just the way he wrote.

This is by Samuel Johnson, in 1765. Note that a "quibble" to Johnson was probably what we mean when we say "pun" today.

A quibble is to Shakespeare, what luminous vapours are to the traveller; he follows it at all adventures, it is sure to lead him out of his way, and sure to engulf him in the mire. It has some malignant power over his mind, and its fascinations are irresistible. Whatever be the dignity or profundity of his disquisition, whether he be enlarging knowledge or exalting affection, whether he be amusing attention with incidents, or enchaining it in suspense, let but a quibble spring up before him, and he leaves his work unfinished. A quibble is the golden apple for which he will always turn aside from his career, or stoop from his elevation. A quibble, poor and barren as it is, gave him such delight, that he was content to purchase it, by the sacrifice of reason, propriety and truth. A quibble was to him the fatal Cleopatra for which he lost the world, and was content to lose it.

Here's the opening sentence of a book by Francis Bacon, for reference. This was published in 1627.

"We sailed from Peru, where we had continued by the space of one whole year, for China and Japan, by the South Sea, taking with us victuals for twelve months; and had good winds from the east, though soft and weak, for five months’ space and more."

It's a run on sentence, but completely comprehensible, right? That's because it's trying to be straightforward prose. Bacon was an incredible author, but he wasn't actively trying to be clever in the way Shakespeare was most of the time. It's about trying to directly convey events, not about the words. Compare that to:

“(She) says that she's finding it especially hard to take when these earnest ravaged folks at the lectern say they're 'Here But For the Grace of God,' except that's not the strange thing she says, because when Gately nods hard and starts to interject about 'It was the same for--' and wants to launch into a fairly standard Boston AA agnostic-soothing riff about the 'God' in the slogan being just shorthand for a totally subjective and up-to-you 'Higher Power' and AA being merely spiritual instead of dogmatically religious, a sort of benign anarchy of subjective spirit, Joelle cuts off his interjection and says that but that her trouble with it is that 'But For the Grace of God' is a subjunctive, a counterfactual, she says, and can make sense only when introducing a conditional clause, like e.g. 'But For the Grace of God I would have died on Molly Notkin's bathroom floor,' so that an indicative transposition like `I'm here But For the Grace of God' is, she says, literally senseless, and regardless of whether she hears it or not it's meaningless, and that the foamy enthusiasm with which these folks can say what in fact means nothing at all makes her want to put her head in a Radarange at the thought that Substances have brought her to the sort of pass where this is the sort of language she has to have Blind Faith in.”

This is, of course, a sentence out of David Foster Wallace's Infinite Jest, and published in 1996. It's not especially easy to read, is it? That's because it's an intentionally-convoluted rant about grammar. Now imagine that you had no idea what AA was and had never heard the phrase "but for the grace of god" and you're getting closer to the meat of the problem.

TL;DR: The KJV was written in an intentionally archaic and poetic manner, but it's still fundamentally a translation that's about conveying ideas to the reader. Shakespeare was being criticized a century and a half after his death for his tendency to forgo coherence for wordplay.