EU member states do not really fall under the control of the European Commission in the way that states are controlled by an executive body. Moreover, sovereign states usually have a foreign policy, which distinguishes it from the semi-independent states or federation states that may fall under it. The CFSP does not over ride the foreign policies of member states-they are able to pursue their own diplomatic relationships. I'd say these are a couple of significant ways in which the EU differs from a sovereign state.
By the way, country to me seems to be an indistinct term used in general situations or by laymen. States or nations (or nation-states) seem to be more commonly used technical terms.
I agree with you thought that the EU blurs the distinctions. It has been said that the nation-state was born in Europe and will die there.