Interestingly, the main purposes of passports is to control who leaves a country and to provide some assurance that you can come back. So the reason countries demand foreigners hold valid passports is to make sure they will be able to leave after their visit and could be easily deported should the need arise. I believe that's also the reason for rules preventing entry to many countries with a soon-to-expire passport.
Without passport, local authorities would typically first need to ascertain their nationality and have the relevant consulate confirm it and issue a laissez-passer. That's why desperate people sometimes try to get rid of their passport or hide it in the hope of defeating immediate deportation (and occasionally also to claim another nationality to get a better shot at asylum).
Therefore, historically, the main reason countries invalidate passports is to prevent people from coming back. For example, during a short time on the night of November 9, 1989, border guards were instructed to invalidate the passport of anybody who wanted to leave the GDR with a special stamp (they quickly gave up and just let people through). In less dramatic situations, people also often have to surrender their passport to make it more difficult to leave, e.g. following a court order.
None of this seems very relevant to Mr. Snowden's situation as he probably does not want to come back to the US for the time being. Invalidating his US passport first and foremost prevents him from doing just that.
However, as long as he holds onto it, it still would look very “valid” to any third country, no matter what US law has to say about it, as there is no legal framework to prevent people from traveling between third countries, no global database of invalid passports and no other mechanism to enforce this supposed “invalidity” (there is an Interpol database of stolen and lost travel documents but it's far from universal).
Once you are stuck somewhere with an expired passport and no hope of getting help from your own country's consular network, your main recourse is to seek another nationality. Failing that, you could try to obtain some sort of travel document from a third country. Several have been created for refugees or stateless people who cannot get a passport from their home countries (Nansen passport, 1951 Convention travel document, 1954 Convention travel document…). They would typically be issued by the country where a person usually resides but, unlike a passport, do not imply that the holder is a citizen. Alternatively, if a country is prepared to grant you entry but not to give you another status right away, you could get a laissez-passer (from the destination country) valid for this one trip.
Of course, none of this really matters for Mr. Snowden. At this stage, it's a safe bet that any decision regarding his fate would be taken at the ministerial level, on a purely political basis.