Well, I think there is no big difference between these two when you try to use them as adjective meaning the best. Context is more important. 最好 is more like a statement, a matter of fact. e.g. 我们公司的焊接技术是整个城市里最好的。If you want to cheer athletes up in a competition or convey your proud for the country, it will sound too smug or arrogant, though maybe sometime it is true. And, yeah, if you use this word before a verb, it can mean "had better do something", but it's not the case here. 最棒 is more informal and emotional. I think it conveys the idea that hope we are the best or I think that we're the best. But if you use 最好，it's more like "you know I'm the best." So, hope this answers your question. Also, it should be 中国是最棒的. Don't forget the 的, the sign of adjective~

First of all, you had better add "的" behind 最好/最棒 to make it more reasonable. (Because 的 is a sign of an adjective). Oh, speak of this, 最好 means "had better". (For example, you'd better do something.你最好做点什么). While 最好的 means "the best", and it has the same meaning with 最棒的. Normally we use 最棒的 in spoken Chinese.

中国“最棒”！is the same meaning with 中国是“最好的”！ 好is good . 棒is great、awesome!