Information Unit Nomenclature: Bits or Shannons?

I must have missed something basic here - is there some reason that "physical data storage capacity", and "information-theoretic information content", should have different units? Aren't they the same thing?

I guess a string of 2^32 '0's would take 4 gigs, but it's information content is much smaller. Is this what he means?

I have heard of people using natural logs instead of base-2 logs to compute the average entropy of a distribution in which case information is measured in "nats" instead of "bits".