Humans across all civilizations and time periods appear to be fascinated with the concept of the "end of the world".
It seems like every generation has a somewhat popular "end of the world" theory and date within their lifetimes that people on the fringe appear to latch onto more fervently, but still remains popular amongst the mainstream.
Does this stem from a peoples' desire to feel "special"? Or something else?
Primarily I'd suspect your suggestion that people have a desire to feel "special" is the main reason behind it.
Beyond that, though, the interests of religious leaders who preach the upcoming end of the world (heads of churches, congregations, religious organizations, etc) are best served if they preach it will happen during their followers' lifetimes. After all, what greater incentive for your followers and those who trust what you're saying to support you, stick to you, and prop you up than the belief their association with you is essential to emerging from the end of the world saved and eternally blessed? Why would they bother listening to you if what you were saying didn't really apply to them, but some future generation that would exist long after they'd died? If someone's going to propose the idea of an upcoming end of the world, they might as well do so proposing it will take place during the lifetimes of those listening to them for any real impact. I think this is at the root of a great deal of the belief in an end of world via a religion's proposed means (such as an apocalypse or rapture, vs. just a big asteroid or something).
Ultimately, though, I think it primarily has to do with people wanting to feel special, and this goes beyond just the religious plots. Their own actions, lives, beliefs and decisions are given greater meaning within the context of serving the aim of their personal benefit in the case of some sort of Rapture or apocalypse, or within the context of their being the last set of actions, lives, beliefs and decisions to ever occur. If the end of the world is only something some unknown, undetermined future generation will have to deal with, it takes away from someone's own significance in the present.