Unity, solidarity, integrity, union can all denote a combining of all the parts, elements, or individuals into an effective whole, or the property or character of the whole achieved by such a combining.
Unity is the comprehensive term applicable to wholes formed either of persons or of things; it may characterize such diverse things as a people, a nation, a church, an association, or a natural or artificial structure (as the human body or a cathedral) or a work of art (as a drama, an epic, a painting, or a bas-relief).
In every case it implies oneness, especially of what is varied or diverse rather than uniform in its elements, that is gained by the interdependence of parts or individuals and by the cooperation of all so that each within its proper limits helps in effecting the end of the whole.
Unity often implies a oneness of spirit that results in a group of persons when there is harmony and concord.
Solidarity denotes a kind of unity in a group (as a class, a community, or an institution) which enables it to show its strength, express its opinion, or exert its influence both through individuals and through the whole with the force of an undivided mass; the term implies unwillingness in individuals or in subgroups to go counter to the interests, aspirations, or will of the group as a whole.
Integrity is used chiefly in reference to wholes that have been built up so that each stands as a thing marked by completeness and a unity dependent on the perfection of its parts and their mutual interdependence; the term usually stresses soundness, undividedness, or freedom from impairment. (see also: Honesty vs Honor vs Integrity vs Probity )
Union is the general term for the act of uniting several things to form a whole or for the body or organization which results from such a uniting. However the term can carry the deeper implications of a thorough integration of parts and of their harmonious cooperation.