William Shakespeare's Macbeth is one of the most tragic of all his plays. Known superstitiously in the theatre world as "the Scottish play" because of so many freakish and deadly incidents during the performances over the years, Macbeth tells of the tragic downfall of a man whose overreaching ambition caused his demise. Of course he had some help along the way, and it is that help (primarily from the witches, of course) that makes Macbeth's story a tragedy rather than just a tale of arrogance and overreaching.
You ask for some comparisons, though I suspect what you really want is some contrasts between the two works: ways in which they are different. Let us assume that most of the story is the same between the original and the modern adaptation to which you refer; it is probably the points of contrast which are most interesting to examine.
The 2004 Scandinavian production was set in Sweden and adapted and directed by Bo Landin and Alex Scherpf. Of course the original was set primarily on the moors and heaths of a rather wild and violent Scotland. It is not surprising, then, that there are some variances between the two plays.
One of the most significant changes in the modern adaptation is the language. Shakespeare's language is both familiar and transformative. He is the first significant writer to use what is now called modern English (though of course some people might call it antiquated). In contrast, the 2004 Macbeth was the first feature-length movie to be filmed in Saami.
Saami is a kind of indigenous, gutteral language which is spoken by very few people (compared to most languages). This is an intriguing choice, as it is a rather rough and harsh-sounding language. Interestingly, this quality makes it perfectly suited to the severity and harshness of this tragic story.
Another point of comparison between the two works is the setting. The modern adaptation is set in a frozen tundra in Sweden, and this is another interesting but appropriate choice. Just as Shakespeare's story is set in the harsh, wild Scottish moors, this adaptation is set in a harsh, bitter, and inexorable world of snow and ice. Additionally, the whiteness of the snow and the colorlessness of the ice is the perfect backdrop to add emphasis to the red blood which is so liberally shed throughout the play.
One last comparison between the two versions of the story is how the witches are portrayed. In Shakespeare's telling, the witches are odd creatures who, though they appear and disappear rather unexpectedly and mysteriously, are generally assumed to be and portrayed as real creatures (women?). Banquo wonders this about them:
That look not like the inhabitants o' the earth,
And yet are on't? Live you? or are you aught
That man may question?
In the modern adaptation, the witches only appear in or through big slabs of ice. We see them as distorted images (think about seeing something through a chunk of ice--you would not be able to be absolutely certain about any of the details you see or even if what you are seeing is what you think you are seeing) rather than actual beings. This visual choice adds to the mystery of the witches and their surprising predictions for both Macbeth and Banquo.
While in one way this depiction of the witches should make Macbeth more hesitant to believe the Weird Sisters, in another way they are more believable as other-worldly creatures who might truly be fortune-tellers or seers. This adds to the tragedy of Macbeth's fall.