The EEAS has an informative piece on the Common Security and Defence Policy. The arguments you ask about can be found by looking at what the policy aims to do: the Petersberg Tasks (quoting from the previous link by EEAS).
These tasks, to me, seem more outgoing, whereas NATO is primarily aimed at defending member states. The reason for doing this through the EU rather than as individual member states is the same as why anything is done through the EU (as opposed to individual member states): because it's more efficient to work together.
The Petersberg tasks formed an integral part of the then European Security and Defence Policy (ESDP) - now Common Security and Defence Policy (CSDP) - and defined the spectrum of military actions/functions that the European Union can undertake in its crisis management operations.
The Petersberg tasks were first agreed upon at the June 1992 Western European Union (WEU) Council of Ministers near Bonn, Germany. Article II.4 of the subsequent ministerial declaration outlined the following three purposes for which military units could be deployed:
The term ‘peacemaking’ was adopted as a consensual solution and as a synonym for ‘peace-enforcement’. The Petersberg tasks were subsequently incorporated into Article 17 of the Treaty of the European Union (TEU) through the Treaty of Amsterdam.
The 2009 Treaty of Lisbon (TEU Art. 42) then further expanded these tasks to include:
- humanitarian and rescue tasks;
- conflict prevention and peace-keeping tasks;
- tasks of combat forces in crisis management, including peacemaking;
- joint disarmament operations;
- military advice and assistance tasks;
- post-conflict stabilisation tasks.