1. Plethora of scriptural citations/references vs. Absolutely none
Pilgrim's Progress is, from the very first page on, literally dotted with either scripture citations or references to scripture citations. This is not merely John Bunyan's opinion or his own personal theology, but the teachings of scripture put into a literary format. Many modern versions of Pilgrim's Progress either place the scriptural index right in the text itself, or at the end of every chapter.
The Shack, by stark contrast, has absolutely no scriptural citations, nor references to scripture. None. There could never be a scripture index because there are no verses to cite. Everything is from the mind of William P. Young, and everything that is written is simply from the substance of William P. Young's personal beliefs. At times scripture is vaguely referred to (such as Mack asking "Weren't you always running around killing people in the Bible?", etc.), but never is a direct reference made.
This plainly reveals the original source material for these two works: John Bunyan's source was the Bible; William P. Young's source material was an idol known as William P. Young's theology. This is probably one of the most important differences between the two.
2. Orthodox Metaphors vs. Heretical Metaphors
Metaphors abound in Pilgrim's Progress, and virtually every character is a representation either of an attitude, a virtue, a vice, or a worldly truth. For example, the character of Pliable, who joins Christian early on but quickly abandons him at the Swamp of Despondence, is a representation of individuals who enjoy the idea of salvation, but then quickly run away at the conviction of their sins. Another example is Faithful, Christian's initial fellow pilgrim who is martyred by the residents of Vanity Fair and becomes a literal representation of the command from scripture to "be faithful until death" (Rev 2:10).
There are, however, very few metaphors in terms of God Himself. Christ is referred to as "Lord of the Hill," "King of the Land," etc., but no one within the story specifically represents Christ in any way other than what He is as the Glorified Lord. There are some who say that the character of Interpreter is meant to be the Holy Spirit, but even if this were so, it would not represent the Holy Spirit in Person so much as what the Holy Spirit does for the believer (that is, endow him with saving knowledge regarding Christ). While Bunyan has fun with literary interpretations, he does not stray too far from what scripture says of a believer's world view and, most of all, scriptural theology.
The Shack, by contrast, is an entire book centered around what is ultimately a bad metaphor for the Trinity (three people inside a shack). To top it off, every ethnic group seems to be represented therein: the Father is an old black woman, the Son is a Middle Eastern man (somewhat fitting, I suppose), and the Holy Spirit is an Asian woman. We are later introduced to a Hispanic woman named Sophia who is said to be the personification of God's wisdom.
What's more, William P. Young's literary interpretations are simply heretical. Papa (the Father character) reveals crucifixion scars on her hands and says, "We were there together," meaning that the Father and Son were crucified together and suffered together, which has been considered heretical since the early days of the church. It likewise contradicts the teachings of scripture that we are reconciled by the Son's body (Col 1:21-22) and that it was the Son who took upon flesh (John 1:14) and humbled Himself before men (Phi 2:5-11). The Father could only have scars if He had a physical body, and unless we are Modalists, we cannot rightfully say that He did.
This is probably one reason why so much of Bunyan's book deals with metaphors pertaining to believers and their interaction with faith and the world, rather than direct interaction with God. Any false representation would have led to dangerous grounds, and most likely Bunyan sought to avoid such traps.
3. Obvious Allegories vs. Near Literal Representations
As already stated, Bunyan's Pilgrim's Progress is an entirely allegorical book. Like Dante Alighieri's Divine Comedy, it was never meant to be a literal interpretation of anything. Bunyan avoids any possibility of this being accepted literally by stating early on that all of it is a dream, and reminds the reader throughout the book that he is simply repeating what he saw in a dream.
Young, on the other hand, opens up The Shack with a claim that his story is true and that there is a real person named Mack who shared this experience with him. Now granted, I know that Young doesn't really believe that Mack exists nor does he claim this in public, but this leads to a different feel for The Shack than Pilgrim's Progress. That is, while Bunyan introduces the story as a dream, and hence the reader understands that the language of Pilgrim's Progress is that of a dream, Young introduces The Shack as a first-hand account, hence readers are supposed to have the mindset that a man named Mack really did go through all these experiences. That is, the reader is led to have a mindset that God really would manifest Himself as He does in The Shack.
If a man named Mack had indeed experienced all that is told of in The Shack, I would have promptly pointed him to Paul's warning that "even Satan disguises himself as an angel of light" (2 Cor 11:14).
4. Absolute Truths vs. Vague Theology
The main protagonists of Pilgrim's Progress (Christian, Faithful and Hopeful) meet various people along the way and, in particularly near the end, hold long discussions with them. What is thoroughly discussed is not only what is wrong and what is right, but that the wrong is an evil wrong that can benefit no man. The character of Ignorance, for example, is told by Christian and Hopeful that he cannot hope to enter the Celestial City (heaven) by his own ways, but he rejects their pleas and is in the end cast out by the Shining Ones (angels) of the city. Another example is seen in the character of Mr. By-ends and his friends, who discuss theology but are thoroughly silenced by Christian and Hopeful when their beliefs are shown to be completely fallacious.
Furthermore, the theology of John Bunyan is plainly seen in the pages of his book. One cannot walk away from Pilgrim's Progress and wonder what Bunyan believed in regards to salvation, our role in salvation, and God's sovereignty. In fact, if one reads Bunyan's purely theological works after reading Pilgrim's Progress, they will find a great consistency with the novel. Everything is there for you, in black and white, and easy to understand. Considering Bunyan wrote this in prison, in the midst of his own persecution, we should not be altogether surprised.
By contrast, The Shack reads like an Emergent book in the sense that the theology is at times so vague that it's difficult to lay down exactly what Young is trying to say. The one that stood out to me the most was William P. Young's Jesus responding to the question "Do all roads lead to you?" with: "Most roads don't lead anywhere. What it does mean is that I will travel any road to find you."
This, of course, leads the book to become very dangerous. When our Blessed Lord speaks of tares and wheat in the gospels, the tares He refers to are plants that look like wheat but are actually poisonous, and cannot be fully shown for what they are until the time of harvest. Young's book, like so many, is essentially a literary tare. It might, from the initial reading, seem orthodox, or even semi-orthodox, but when one really gets down and studies what he is saying, one realizes just how heretical it is. One of the biggest examples of this in the book (next to the previously mentioned teaching of universalism) is the teaching of a restorative hell, shown in the scene where Mack reconciles with his deceased father. Young does not flat out say he believes in a restorative hell in the book, but when one reads through what he is teaching, that is the obvious conclusion.
Bunyan is not afraid to come out and say, "This is what I believe, it is based on the word of God, and I am sticking by it whether you like it or not." Anyone afraid to do likewise, and anyone who feels they need to essentially hide their theology, should themselves and their work be considered very dangerous.
5. God-centered Goals vs. Man-centered Goals
The goal of Bunyan's Pilgrim's Progress was to be an evangelical tool. It was written to explain the gospel in literature. It was written, first and foremost, to present God's message to the world while remaining faithful to the word of God.
The Shack, on the other hand, is essentially something to make someone feel better. Young himself experienced some great suffering in his life, and many who hold The Shack to be a wonderful piece of literature are likewise those who suffered from some tragedy in their life. Young has even said in interviews and lectures that certain representations (such as the crucifixion scars on the Father) were done to make people feel closer to God. The entire goal of the book, then, is to please men through emotional connections.
Some might wonder why I am presenting this as a contention against Bunyan. After all, isn't it good to make people feel better? I respond that the idiom is "the road to hell is paved with good intentions," not "the road to heaven." There is nothing worse than presenting falsehood and distortion of truth in the form of spiritual healing. It is just as bad as a doctor giving poison to a patient on the pretext that it will save their lives from a disease.
As such, what Young did in The Shack was try to make people feel better by completely distorting scriptural teachings and misrepresenting how God has revealed Himself to be. Unfortunately many people, seeking to please their itching ears, have grasped onto his work as a great piece of literature simply because it makes them feel better. I for one would much rather have a terrible life on earth with hopes of a glorious life in heaven than a great life on earth with an eternity in hell.