* This topic deals with the Dwarf Mode, not DF in general, as the Adventure mode is very dissimilar to Minecraft *
I had at one point been one of the people who thought that Minecraft was just an attempt to rip off Dwarf Fortress. Eventually, after reading the TVTropes page, I decided to just give it a shot, to humor myself at the least, to be able to come up with more convincing arguments relating to how MC copies DF.
I got addicted. So sue me. (Don't really).
MC is, to an extent, the application of DF to an individual verses a population. DF has you controlling a hoard of dwarves, while MC has you controlling an individual. Where dwarves fail to express careful detail (Urist McMiner cancels live: interrupted by ceiling), you can succeed in MC. However, where individuals have slow progress (the digging of massive projects), a population can succeed greatly.
DF has a much greater variety of ores than MC does, as well as a much deeper and realistic (until you hit slade) composition. Vanilla MC only has iron, gold, diamonds, and emeralds; the lattermost serving no purpose aside from currency.
MC has a much gentler learning curve if you have Wiki access, or almost the same without; this is due to the crafting system, and you having to know how everything is crafted. While the system itself is intuitive when you get to know it, some receipes are downright confusing unless you have built it before. DF's biggest hurdle IMO is the interface.
Both games are designed to be modded; going by what I heard about Masterpiece, and my experience with Tekkit Lite, I say that the two are approximatly equal in this respect.
DF's goal is to manage a population of dwarves into doing whatever you want them to do, from building megaprojects to just surviving day after day. MC's goal is to permit creativity and adventure. With the practically infinite landscape availible in MC, you will never run out of resources. But the restricted area in DF means that eventually you will have no more material from the ground that you can take, and you can't go out to get more. This is one of the challenges you have to overcome in DF.
MC has an end winning condition, and an optional boss. DF only has the optional boss, but it is easy to accidentially trigger it.
The most major point that people debate over is that both worlds are composed of blocks and foster underground development. I can say, from lots of experience in both games, that they are both composed of blocks, but MC doesn't entirely foster development underground. MC seems to be about exploration, where you have lots to explore both above the ground and below, while DF seems to be about development, making do with a fixed area and making the most of it. While you can do this in MC (not as efficiently as in DF, however), you can not do the adventuring component directly in DF; another point for MC comes from how you can go directly from adventuring to development and vice versa, while you can not do this in DF. However, DF makes up for this by offering a view into the history of the world through artifacts and other pieces of art. MC doesn't have such a history.
MC costs money, which gives you an unlimited landscape for you to shape. DF is free, and gives you an army of dwarves to do your bidding. While there are similarities, there are also major differences. Differences which contribute a major, major change in gameplay.
Also, Notch (the original only dev of MC, and only dev for quite some time; has since stepped down) has stated that Minecraft was intended to be similar to DF, but with a different feel. And while the game, in a few core mechanics, feels very similar, the development veered off the road of becoming a DF clone, much as the threads here stray from their original topic.
tl;dr: MC and DF are different games with different gameplay revolving around the concept of surviving in an enviorment you can shape as you feel fit.
edit: fixed a mistake