ANSWER: One means "poor," and the other means "really poor." Also, one of them can also mean "awkward."
My Korean-English dictionary does not really show much difference between 궁색 and 군색, but my Korean-Korean dictionary defines 궁색 as "아주 가난함," which translates as "very poor," while defining 군색 as "필요한 것이 없거나 모자라서 어렵고 답답함," which seems to essentially translate as "needy." Also, my Korean-Korean dictionary shows a second definition for 군색, which is "자유롭거나 자연스럽지 못하여 거북하고 어색함," and that essentially translates as "awkward."
Anyway, when Koreans talk about someone being "poor," they seem to use 궁색하다 much more than 군색하다, but when they talk about "an awkward excuse" or "a lame excuse," originally the only correct phrase was "군색한 변명," not "궁색한 변명."
Donga's Prime Korean-English Dictionary (1998 edition)
