Minidisc is nice, and I strongly considered it... before realizing how superior the iPod was in many ways.
I like to have a wide variety of music with me at any time. I'm also a professional, which means I need to have a stupid amount of things in my pockets - cell phone, PDA, wallet, keys at the very least. I can't keep them in the same pocket because it's too bulky and things will get scratched. What this means is that I can barely fit in a music player, and I definitely couldn't fit in a bunch of minidiscs. Strike 1.
Even if I could fit a bunch of minidiscs, how many would I need to carry around? That's a lot of extra shit to look after. You can get 3 hours onto your average minidisc. Okay. The music I listen to, that's less than 3 albums. But we'll call it 3. Say I feel like carrying around 10 minidiscs, which I never would - that's 30 "CDs" of music. That's a fair amount, but that's way too much to carry. I can put 200 CDs of music on a 10GB iPod. Strike 2.
It's faster to put music onto my iPod. I can grab a thousand songs from my playlist in the morning, drag them onto my iPod, hop in the shower and by the time I'm toweling off my sack, they're all there. I might not listen to a thousand songs in a day, but it gives me a great variety. iPod is, to me, more convenient than minidisc. Strike 3.
The cost of an iPod is what you pay up front - that's it. No extra batteries, no extra discs. Buy it and you've spent all you will.
The battery life is also quite good. I get 12 hours easy on a charge. You might scoff at that, and you'd be mostly right. I can't change the batteries... but I never need to. I go home, plug it in and an hour later, it's back at 80% capacity. It takes 3 hours to charge fully. You should sleep more than 3 hours a night. So there you go.
Minidiscs are pretty cool, but I definitely think that there are some serious advantages to hard drive-base MP3 players. My player of choice is the iPod, but others are similar. If the goal is simply "play music anywhere", it's kind of hard to beat it. It's got good battery life, it's light, it looks good and it fits into pockets easily.