http://nixos.org/nix/ summarizes some of the advantages of this approach. To summarize, it promises to be fundamentally safer and more flexible than traditional package managers. Package management, IMO, is an area where a "purely functional" approach could really shine, since correctness and replicability are so important.
Well, the obvious answer is that it is an advantage for the same reason that being purely functional is an advantage in other situations.
More specifically, the benefit of having no side effect in the package management is that it greatly simplifies the management of multiple versions of a given package. It also simplifies the rollback of an unwanted operation in most cases.