国家 and 国 both mean "country" or "nation". However, what's the significance of the character 家 in the word? Does it really add the meaning of "home" or "family"? Is there a cultural or historical basis for this? Or did the words and/or characters mean different things in the past?
Have a look at http://zh.wiktionary.org/zh-hant/%E5%9C%8B%E5%AE%B6, which explains lots of traditional meanings of 国家.
The basic meaning is related to
When Chinese and Japanese were finally exposed to the Western world in 19th century, they had to create new words or re-use old words to translate the concepts and notions which were not existed in their traditional culture. Eventually