First of all, lots of good stuff has been posted here so far. My thoughts on this are right in line with a lot of what is already here. So basically, +1.
Still, your question made me journal a lot of thoughts down today! So I'll share some of them. This goes back to perhaps what is one of the greatest mysteries of our faith...
Like many have said here, it all comes back to Problem #1... the very reason why we are all here. The fall of man.
Also, like many have quoted here, the first verse that I thought of was 1st Corinthians 10:13
No temptation has overtaken you that is not common to man. God is faithful, and he will not let you be tempted beyond your ability, but with the temptation he will also provide the way of escape, that you may be able to endure it.
Clearly, the key here is that the promise from God is that we will be given all that we need to ENDURE temptations. Note how the Word does not say eliminate them. But, as you asked, why is this the case?
Here's a quote from John Gill's bible commentary that I found particularly useful, about the last phrase: "that you be able to endure it"
"for God does not always think fit to remove at once an affliction or temptation, though at the earnest request of his people, as in the case of Paul, ( 2 Corinthians 12:7 2 Corinthians 12:8 ) yet he gives them grace sufficient to endure and stand up under it, yea, to get the victory of it, to be more that (than? typo here?) conquerors, and triumph over it."
As Gill indicates, perhaps the persistence of desire then, is an existential question... without the temptation of sin remaining extant, then we cannot actually triumph over it.
But why is a victory over sin important? Simply put, a conscious choice to renounce our body's sinful nature and to live in the ways of the Lord is necessary, if we are to be reunited in perfect communion with him in Heaven. As we know from the fall, sin and God cannot coexist.
Fair warning, I'm going to comment a bit further away from the scriptures here, so take this with a grain of salt-
In a sense, the fall of man introduced the culpability of sin into our existence. Before obtaining the knowledge of good and evil, we actually had the ability to sin, but yet we never did. Why is that? Foremost, we did not have the desire to sin, but secondly, Adam and Eve originally lived perfectly within the ways of the lord, which did not lead them into sin (unknowingly). They were but like children, blameless.
As human nature dictates, along with the knowledge of sin, comes a temptation to sin. Perhaps we never went beyond obtaining the knowledge of good and evil, to obtaining the knowledge of "life and death." That is to say, the understanding of the beauty of righteousness, and a full understanding of the consequences of choosing one over another. In other words, why truth stands on its own, and triumphs over evil. Such a knowledge must then be gained through experience and the discernment of our own minds. However, God, through his mercy, has given us a shortcut to obtaining the essence of this knowledge: the Word. Still, we are weak and forgetful, and will most likely still chose to sin against him, as many of us, myself included, have done.
Although we can now knowingly sin against God the Father, the singular act of Jesus's sacrifice is enough to indemnify us from such an iniquity. Now that's powerful. Let's never stop thanking God for his grace and mercy!