First of all, the Council of Europe and the European Union are two separate organisations.
The Council of Europe was created in 1949 by ten Western European states. The Statute of the Council of Europe excludes matters relating to national defence, but is otherwise broad in scope (Article 1):
The aim of the Council of Europe is to achieve a greater unity between its members for the purpose of safeguarding and realising the ideals and principles which are their common heritage and facilitating their economic and social progress.
This aim shall be pursued through the organs of the Council by discussion of questions of common concern and by agreements and common action in economic, social, cultural, scientific, legal and administrative matters and in the maintenance and further realisation of human rights and fundamental freedoms.
Article 3 of the Statute lists the basic criteria for membership:
Every member of the Council of Europe must accept the principles of the rule of law and the enjoyment by all persons within its jurisdiction of human rights and fundamental freedoms, and collaborate sincerely and effectively in the realisation of the aim of the Council as specified in Chapter I.
Democracy, human rights and the rule of law have become the core competencies of the Council of Europe.
The Council of Europe is an inter-governmental organisation, where the decisions are taken by the Committee of Ministers (representatives of governments). The organisation has been able to conclude a large number of Conventions (treaties) between the member states in a wide range of fields covering both human rights and other questions of common interest. The Committee of Ministers, assisted by the Secretary General, the Commissioner for Human Rights and specialist committees, monitors member states’ compliance with these Conventions and issues recommendations. This advisory and monitoring work is pursued in the inter-governmental sphere, largely outside public scrutiny.
Although it fell far short of the aspirations of the European federalists, the Council of Europe was attributed a Consultative Assembly, a deliberative organ with an advisory role. Nowadays known as the Parliamentary Assembly it gives the organisation some stature above strict inter-governmental co-operation. The members of the Assembly are chosen by the national Parliaments. It has been called a para-legislative body.
The most remarkable step outside the inter-governmental sphere has been the evolving Convention for the Protection of Human Rights and Fundamental Freedoms, especially its control mechanism crowned by the supra-national European Court of Human Rights, which exercises judicial control leading to binding judgments.
Today the geographical scope of the Council of Europe is pan-European, encompassing 800 million individuals and 46 member states. The new state of Montenegro has applied for membership. Only Belarus is beyond the pale.
Of these members, 27 are also members of the European Union. In addition, the organisations share values and are active in the same or related fields.
Overlapping, duplication, competition or co-operation? There is, to say the least, cause for clarification leading to complementarity and synergy.