Because of software and hardware choices made by Google and Samsung, the Galaxy Nexus can't support USB Mass Storage for transferring files to and from a computer. However, the issue doesn't extend to all other devices running Android 4.0, as Android engineer Dan Morrill confirmed in a comment on Reddit today. Since Android 3.0, device manufacturers have had the option to create "one happy volume" for storing both apps and media instead of creating two, separate partitions. This definitely makes for a better experience on the phone, but the tradeoff is that transferring data could be a little more complicated for some users.
Other devices on Android 4.0 that use removable SD cards for storage will continue to be mountable on host PCs, as the media remains separate from the OS. Galaxy Nexus users with Windows PCs shouldn't find the feature's loss to be too much of a problem, as the OS natively supports the MTP standard the phone uses to transfer files. Mac users, however, will have to use the Android File Transfer app. Check the source for a more detailed explanation of the issue.