Belegbeth makes a good point about the distinction between weak magic and
rare magic.
I think a rare magic campaign will always fall into the same trap as a regular
D&D campaign, however, because magic for the PCs isn't rare. The PCs will
have access to powerful spells and a wide array of powers, making it very easy
for them to overcome obstacles that are traditionally used as plot hooks. The
DMG has some good advice on this subject, suggesting that you create adventures
*requiring* the PCs advances powers, rather than nerfing them to force things
down to a more familiar milieu. The King's assassin was an extraplanar
creature with links to an earthly secret society, and the PCs need to use several
powerful spells just to find this out.
A weak magic campaign, on the other hand, makes it much easier to create mundane challenges (survival in the wilderness, for example, can become a major challenge if
magic can't create food and water out of thin air). The setting is more familiar, the
stakes easier to empathize with. This is attractive, but it can get dull at higher levels because the abilities of your character don't change as dramatically.
It's a tough choice, but there's a third option: Limited Magic. The main reason that D&D has so much trouble simulating our favorite fantasy worlds is that their authors, high or low magic, generally placed limits on what magic can and can't do! Maybe wizards needed extensive time and bizzare components to make any spell at all work. Maybe the effect was limited by the magician's personal energy. Maybe magic couldn't create any effects that were blatantly unmagical.
The problem with D&D is that there's a spell for any occasion, and 99% of them have no real cost to the caster. 1st edition at least had those aging rules for powerful spells that severely limited your ability to get them from NPC's! What's worse, wizards and clerics can invent new spells at whim. If magic can do anything, what's the challenge? It would have been great if the PH had a couple of pages explaining what magic *is*, what it's good at doing, and what it has trouble accomplishing. As it is, it has no limits. But then, of course, it wouldn't be "generic fantasy".
So for those of you having trouble getting high-magic campaigns with enough drama, try designing a campaign with some specific rules on how magic works. Place some restrictions on it. Chain it down under the cultural bias of specific regions in your world. Go through the spell list and ruthlessly cull the offending spells. Restrict development of new spells the same way. Then you can make magic into a powerful tool, but not one that's right for every occasion. You can allow players to get 9th-level spells, but you don't need to give them Wish.
Personally, I get a real kick out of the low, low-magic levels. I find that players are more ingeneous when they have very limited options, and I like the sense of caution. But I also like to see my character growing and developing in ways that aren't strictly mundane, acquiring funky powers or undergoing strange transformations. Maybe even getting to play with a powerful artifact for a session or two before the plot tears it out of my sweaty little hands.

C'mon, gimme the best of both worlds!
--Ben