The biggest difference is with Polonius and Ophelia regarding Hamlet's madness.
In the actual play (Act II.i), Ophelia describes to her father:
Lord Hamlet, with his doublet all unbraced;
No hat upon his head; his stockings foul'd,
Ungarter'd, and down-gyved to his ancle;
Pale as his shirt; his knees knocking each other;
And with a look so piteous in purport
As if he had been loosed out of hell
To speak of horrors,--he comes before me.
This is knows as the "silent interview" (between Hamlet and Ophelia), in that Hamlet does not speak. Later, he gives her a note. Even though he doesn't speak, he appears to be mad (crazy).
In Act II of the Zeffirelli version (1990), Polonius spies on Hamlet as the silent interview happens. The audience also sees the silent interview live rather than in retrospect from Ophelia's retelling. So, Polonius and the audience see firsthand signs of Hamlet's madness rather than as a secondary source after the fact.
In film dialogue is better if it is fleshed out live, not delivered in summary apart from its corresponding action. Zeffirelli wants Polonius to be a voyeur from the start. The director wants us, along with Polonius, to see Hamlet's feigned madness. He also wants to give as much screentime to his star, Mel Gibson, as he can.
I totally agree with "mstultz72" about the difference in regards to madness, but I would be amiss if I didn't mention my absolute favorite difference from my absolute favorite movie version of Hamlet EVER: humor! The director, and more specifically, Mel Gibson himself, insert humor where I am not sure Shakespeare meant for there to be. Let me give one quick example. (And by the way, this example was my first inkling into this grand difference that I determined back when I was in high school, which shows how old I was when this movie came out.)
Take the scene where Polonius confronts Hamlet in the castle (Act 2, Scene 2). First, Shakespeare's words don't surprise us as Hamlet says:
Let her not walk i' th' sun. Conception is a blessing, but as your daughter may conceive, friend, look to 't. (2.2.185-187)
Here Mel Gibson plays upon the "sun," meaning the king and upon "conception," meaning both understanding and pregnancy. He is implying, of course, that the King could get unsuspecting women (Hamlet's mother) pregnant as Hamlet himself could get Ophelia herself pregnant. But this is typical Shakespearean humor. What is different is within the next few lines:
Polonius: What do you read, my lord?
Hamlet: Words, words, words. (2.2.194)
As a high school student, I would read Hamlet's reply quite flatly. There is simply no indication of reading it otherwise. No play on words. Nothing. Mel Gibson, however, takes this simple repetition and gives each word a different flair. Mel Gibson's Hamlet looks in the book as if wondering what's in there saying "words, words . . ." and then turns to Polonius and matter-of-factly declares, "words!" Ha! In fact, Mel Gibson reads it in a way that actually demeans Polonius(which is Hamlet's full intention in the scene anyway). As if Polonius is so very stupid and such a fool that he wouldn't even know that words are contained in a book.
I would argue that the way this movie is both directed and acted actually takes Shakespeare's literal attempt at humor and adds to it while not taking away from the author's original meaning. This is the core of why I am so very enamored of this version. It remains my legitimate favorite (with my "illegitimate" favorite being the Mystery Science Theater 3000 version).