Librarians. These are people trained in research, in separating the wheat from the chaff. Now, part of my utter faith in the need for librarians is the fact that, well, my best friend is one (or anyway, works in a library; to truly be a librarian, you need a Masters in Library Science). However, I know that she can take the stupidest, most poorly-worded question you can imagine and eventually get the answer the person is looking for. It is very difficult to use a search engine if you aren't quite sure what terms you should be looking for. Eventually, I suppose it's possible to put librarian features on websites, but you'd still need the actual person; it's not a process that can be automated.
Rare book rooms. Libraries provide important tools for the storage and handling of rare books that would not otherwise be able to be in public hands.
Storytime for small children. Sure, you can do audio books, but for one thing, they're awfully expensive. For another, storytime also provides tools for developing social interaction.
Oversized books. There is, at the library where my friend works, a positively enormous book of photos from space. I'm sure you can get the photos online somewhere. However, unless you have an awfully big monitor, you can't get the full impact of seeing the photos in all their glory. (It also includes essays by various astronauts, but words on a screen would convey that equally well.)
From a purely aesthetic sense, there is to me a great joy in just wandering the stacks, even if I'm not looking for anything in particular--which is, now I think about it, another benefit of libraries. Browsing. You can just wander aimlessly among the shelves, picking up a book and flipping through it, running your fingers along the spines, looking at a book because the title or the cover caught your eye. Following Wikipedia links can be fun, but it's not at all the same thing.
"Now everyone was giving her that kind of look UFOlogists get when they suddenly say, 'Hey, if you shade your eyes you can see it is just a flock of geese after all.'"
"You can't erase icing."
"I can't believe it doesn't work! I found it on the internet, man!"