The term "The States" to refer to The United States of America is almost exclusively used by people from outside of The United States, or sometimes by Americans trying to appear worldly.
Like most terms used for a geographic entity that are not used by the residents themselves, the term "The States" is perceived by me (and probably by most Americans) as somewhere between incongruous and derogatory. Although it may be reading too much into it, I think one of the reasons the term is used is because the rest of the world may find it slightly chauvisnistic that citizens of The United States should take the words "America" and "Americans" upon themselves, since truly, anyone in both the continents of the New World can refer to themselves as "Americans".
Historically, the United States of America got its name when it was the only independent nation on the American continents. To refer to ourselves by our geographic location instead of our system of government seems normal. After all, residents of The Republic, The Federal Republic and The Kingdom would hardly like their countries to be referred to by their governmental arrangements, if they could even make any sense of what these names referred to.
It should also be noted that of all the countries in the Western Hemisphere, only the United States of America has the word "America" in its name. Thus the term "The States" is ironic as a specifier, since it is less specific than "America". While only the United States of America is The United States, North America's other sovereign nation is also "The United States" (of Mexico).
So, for reasons of accuracy, as well as as politeness, the term "The States" is one that should be avoided when speaking to Americans.