Recently, I was discussing media use patterns with Erin Reilly, the research director on Project NML. One of the things we started talking about was what a modern library user does.
I'm sure that many people use libraries in many different ways, but for me, a library mostly functions as an aesthetic space. I like to go to beautiful and monumental libraries because they inspire me and make me think about how noble the pursuit of knowledge is.

The Library of Congress foyer, by sandcastlematt.
I like to go to small, stuffy, hot libraries and work in a carrel because I associate carrels with being a serious student: I never feel like I am
really writing a term paper unless I'm writing it in a carrel.

A library carrel, by audreym.
I like to walk through the stacks of a public library because I am more likely to pick up a book I've never read before if I get the chance to pick it up, look at the cover, and turn it over in my hands.
The shelves of the Bird City Library, by Adventures in LIbrarianship.
So the idea of a physical library is pretty important to me, but not for the reasons that one might expect. I spend a lot of time in libraries, but unless I'm just trying to kill time, I don't actually go to the stacks and look for books to check out very often. Instead, I use online catalogs as if they were Netflix. If I want to read a specific book or books on a specific topic, I go online and search for them in the catalogs of all the libraries near me. Then, I order them and just go pick them up when they're ready. I use the "search" function in very specific ways, to find the particular information I need.
But what about the Learning Library? I envision the Learning Library as a tool that includes both aspects of a library: a tool where you can either browse around and have fun or where you can look for a very specific skill that you want to learn about and zero right in on it. I also hope that the Learning Library will be an inspiring virtual place. That might be a lofty goal, but I think it's an achievable one. One of the goals of the Learning Library is to emphasize how much everyone can create and achieve in our participatory culture. I think that's a pretty inspiring idea.
So I'm going to try and keep the idea of a physical library in my head as I work on the not-so-physical Learning Library. I wonder what my children will think of when they think of the word "library" - will they enjoy looking around in the stacks? Will they even know what a carrel is? Or will they associate the word "library" with projects like the Learning Library? I suppose that only time will tell.
"So the idea of a physical library is pretty important to me, but not for the reasons that one might expect."
Beyond the question of associating digital libraries with the idea of a library--but getting to another of your points--to a lot of people libraries are basically churches. Local libraries are chapels, and something like the Library of Congress isn't much different than a cathedral. Online libraries can get at the community feel, but I wonder if they'll ever approximate the grandeur--the worship of knowledge--that something like the Library of Congress evokes.
I'm totally biased though. I got engaged at the LOC.
Andrew, that's awesome. When I was living in DC, I used to go down to the LoC every so often for just the reasons you mention – the sheer magnitude and grandeur of this temple to human knowledge.
Personally speaking, I somehow disconnected from the idea of borrowing books from libraries a long time ago, probably a result of the one-two-three combo of the libraries in my small hometown being seriously understocked; my being notoriously impatient, picky and temperamental when it comes to what I want to read right now; and my desire to build a temple of my own. Now I'm just struggling with the notion of how to keep track of all the media I have.
And that's another issue, isn't it? What does a modern library have to include? Is a library considered understocked if I can't go in and borrow a DVD of THE MALTESE FALCON? What if I can't go in and borrow a copy of THE LEGEND OF ZELDA for the Super NES? What about copies of old radio serials like THE SHADOW? It's obvious that there are some lines that should probably be drawn, as evidenced by the latest version of Delicious Library (a piece of cataloging software for the Mac) including /tools/, for Chrissakes) but where those lines actually are is increasingly fuzzy. Add to this the idea that Morning Edition proposed on NPR yesterday ( http://www.npr.org/templates/story/story.php?storyId=94884967 ) that video games might be the nickel arcade of the Next Depression and the role of a wide-ranging public media library becomes increasingly valuable; as Anne Herbert famously said in the Whole Earth Catalog, "Libraries will get you through times of no money better than money will get you through times of no libraries."
Oh man Delicious Library. When I got a Macbook last Christmas, I basically retired from the world for a weekend just to use the iSight camera to scan my books' barcodes into Delicious Library. Then I exported everything and added it to Goodreads. What's interesting w/r/t to Geoff's comment above is that other people's personal collections are now my library. (And if you've ever used Bookmooch, you know that's literally true.)
What's especially bittersweet is the loss of a physical public space where people can seek out answers to the tough questions, knowing that everyone around them is doing the same thing. Watching an opening shot from Wim Wenders' film Wings of Desire really drives this home: http://www.youtube.com/watch?v=Wi8sYY0pCdE