It's a number of things...
No use in denying it, Apple has done a good job of marketing the iPod and helped make it "the thing" to have, but it seems to be doing better than "fad" -- it seems to have more staying power, so lets look at some of the reasons why.
Hardware Design: The iPod feels good in your hand. It's a solid device. It feels expensive because it is.
User Interface: I won't say it's THE best, because that's always open for debate, but the iPod has got one of the best UIs out there. It's well laid out and the click-wheel matches the UI perfectly.
Sync: Syncing with iTunes is effortless, on PC or Mac.
Entire library -- to go: You say you don't understand why we carry a player with more capacity than battery. That's not the point. The battery lasts acceptably long {for me anyway, and most other people with an iPod} and my whole music collection is there, in a logically organized fashon. If I want classical, jazz, oldies, rock, pop, muzak, it's all there. U2? John Mayer? Oscar Peterson? I don't have to plan ahead what I want to listen to, I just click click click and I've got a playlist.
Quality: This one is debatable. I personally have been very impressed with the files I have ripped to AAC and can hear the difference in quality vs. the same song ripped at the same bitrate for MP3. I could get the same audio quality from MP3, but it would take larger files. I haven't heard great things about the quality of Artac. Weather that is true or a myth? I don't know since I don't have any personal experience with Atrac files. The fact that Sony {used to} made/makes me convert my entire collection to ATRAC before it can go on MD sucks though. My iPod will do WAV, AIFF, MP3, AAC, Apple Lossless, and probably a couple other formatts. No Oggs. There are lots of myths out there about Macs, different cars, even the iPod itself. I personally am willing to accept the idea that the supposed horrid quality of Atrac is a myth, but since I've got no personal experience with it, and since audio quality is such a subjective thing, I personally will reserve my final judgement until I've heard it. Unfortunately, not everybody thinks that way and so I'm sure some MiniDisc sales have been lost on that myth. How many sales did the Newton loose because of myths of bad handwriting recognition? How many sales did Ford loose on the Explorer on the myth that all of them were equipped with faulty ties?
No design is perfect. Not iPod, not MiniDisk, not anything. So lets look at some of the flaws of the iPod.
Battery: Almost a moot point for me, but it's only fair to bring it up. Sure enough, you have to pop the back off to replace the battery. The battery has plenty of life for me and when it does eventually die replacements are reasonably priced. It's a tradeoff, and I'm willing to have the battery sealed inside for the absolute ridigity the iPod provides.
DRMed tunes: The iPod only works with the iTunes music store. This, again, is a moot point for me since the *only* online music store I can use is the iTunes music store, and I think it's the best out there. But if you want to use some other music store, iPod isn't for you.
Also, while some other MP3 players will work with Mac, the Sony ones -- to my knowledge -- will not. At all.
So that's my analysis of why the iPod is preferred so much over MiniDisc -- Apple's great design, their marketing team, the fact that most people who get an iPod truly find it to be a great product, and the fact that for a long time Sony was forcing you to convert all your tunes to Atrac before you could even move them to MiniDisc.