UPDATE (27 July 2012):
Link to Wikipedia article has been updated to reflect the version of the article that was current when this answer was posted. Please note that the current Wikipedia article has been extensively revised!
Well, according to an older version of a Wikipedia article on NoSQL:
NoSQL is a movement promoting a
loosely defined class of
non-relational data stores that break
with a long history of relational
databases and ACID guarantees.
and also:
The name was an attempt to describe
the emergence of a growing number of
non-relational, distributed data
stores that often did not attempt to
provide ACID guarantees.
and
NoSQL systems often provide weak
consistency guarantees such as
eventual consistency and transactions
restricted to single data items, even
though one can impose full ACID
guarantees by adding a supplementary
middleware layer.
So, in a nutshell, I'd say that one of the main benefits of a "NoSQL" data store is its distinct lack of ACID properties. Furthermore, IMHO, the more one tries to implement and enforce ACID properties, the further away from the "spirit" of a "NoSQL" data store you get, and the closer to a "true" RDBMS you get (relatively speaking, of course).
However, all that said, "NoSQL" is a very vague term and is open to individual interpretations, and depends heavily upon just how much of a purist viewpoint you have. For example, most modern-day RDBMS systems don't actually adhere to all of Edgar F. Codd's 12 rules of his relation model!
Taking a pragmatic approach, it would appear that Apache's CouchDB comes closest to embodying both ACID-compliance whilst retaining loosely-coupled, non-relational "NoSQL" mentality.
