Anvilicious/Some Anvils Need to Be Dropped: Very and deliberately so, and actually the reason it became popular: its Christian message is so anvilicious that the Puritans approved of it, deeming it almost as suitable for reading on Sundays as the Bible itself.
One of the characters Christian comes across is Ignorance; a pleasant, friendly young man who tries to get into the Celestial City by doing good works and living his life according to Jesus' example, rather than simply through faith. He gets thrown into Hell for his arrogance.
Misaimed Fandom: Of a more Insult Backfire type of way. Muckraker in the book is a character who is so obsessed with cleaning up mud on the ground that he fails to notice a golden crown above him. Journalists in the early 20th century who investigated big corporations were dubbed "muckrakers" as an insult, but it eventually turned into a positive term thanks to Theodore Roosevelt's speech "The Man With the Muck-Rake", which notes the irony of the nickname. There's also the magazine company "Vanity Fair". If they took the name from Pilgrim's Progress (or were aware of the book when they chose the name), they either don't know the book, forgot or disregarded that the Vanity Fair in the book is a bad thing.
Seinfeld Is Unfunny: The Pilgrim's Progress can be a difficult read for some readers today, including Christians, who aren't able or willing to look past the Values Dissonance, directness and general preachiness of the book. The language doesn't help either, although there are editions with more modern writing.
Values Dissonance: There is some, especially to modern readers due to the increasing secularization of society.
"Abandon your wife and children" isn't generally considered acceptable even as an allegory these days. Also, Ignorance's fate can come off as unnecessarily harsh and arbitrary. ("Look, sorry, you really need to go back and follow the proper path. You've done nothing wrong, so we'll hold a spot for you.")
Christian and Faithful informed Ignorance of the correct way to go but he did not heed the fact.
In the second part, all four of Christiana's sons get married within a span of a few pages despite having no romance arcs with any of their wives. To modern viewers, this would seem like an extreme case of Strangled by the Red String. In the 17th Century, getting married and having a family is seen as more important than experiencing romance, and people in those days usually get married before they fall in love with their spouses, rather than the other way around.
Values Resonance: The second part of Pilgrim's Progress; the story of Christian's wife and children has them take up arms and help people including the mentally-challenged. This affirms that God is available for all and women, children and the mentally challenged can be brave pilgrims as well. This was written in the 17th century, when society had values such as throwing the mentally challenged in a Bedlam House, and most women were expected to Stay in the Kitchen or make babies.