What is the difference between Shushan and Shushan Habirah or is there one, and when does the Meggilah use each one? (Please quote sources)
Ibn Ezra (throughout the Megillah, and also to Gen. 14:1) says that "birah" means a palace, and thus that Shushan Habirah means "the royal palace in the Shushan area." He also says (commentary to Esther 1:1 and 9:2) that Shushan was a province ("medinah") containing numerous cities, most of whose inhabitants were Jews.
[Perhaps, too - although he doesn't spell this out - this might explain why indeed the Megillah, even though it describes the celebrations in Shushan on the 15th of Adar, never explicitly says that this should be done in future years (it mentions the 15th in 9:19,21, but without giving a placename). According to this explanation, that Shushan is the name of a province, then you couldn't correctly say that "in Shushan" they should celebrate on the 15th, because that would apply only to some parts of the province - the walled cities.]
Malbim (to Esther 9:13) similarly explains that the city of Shushan contained two districts, a residential area called just "Shushan," and the royal compound called "Shushan Habirah."
As for msh210's question about "Shushan" in smichus form: maybe it could fit with either of these explanations, in that people would have made a distinction between "Shushan of the palace" and "Shushan" proper.