I'm wondering which of these two expressions (emblem or coat of arms) is used while speaking about the symbol of a particular country.
THX a lot
It all depends upon which symbol you are talking about. Any symbol or animal can be "emblematic" of a nation. Coats of arms are more official.
For example, any depiction of an eagle can be meant to refer to the United States. But this:
is the official coat of arms of the US. (The obverse of the Great Seal of the United States.)
Great Seal of the United States - Wikipedia, the free encyclopedia