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
古代諸侯的封地稱國，大夫的封地稱家. In the classical age of China, there were vassal states and the land of a vassal was called 国. There were also officials titled 大夫 who were given land as well, and a 大夫's land is called 家. Later 国家 was used to mean 国 in some situations (for rhythmic reasons, I suppose). The meaning also got extended to mean the court, or the emperor, or the capital. None of these was today's concept of 国家.
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
国家 was picked to be the translation of the modern concept of a state.