For me, it seems that the main differences between the TNA and Organizational Diagnosis are the scope and methodology. TNA is more about discovering the most important competencies and best practices for a specific role, and defining what are the critical tasks and performance requirements for that role. This can then be used to create a training program for that role.
Organizational Diagnosis may be used as a methodology for more specific objectives, but as a whole it is far broader in scope and involves the processes, strategy, relationships, systems, environment, culture, etc. of an entire organization and involves all members of that client organization. The analysis is more of an ongoing, collaborative process that affects multiple roles (i.e. the entire organization).
Organizational Diagnosis also requires more of a facilitator role that empowers the organization to take ownership of its own solutions and enact them.
Both processes, however, involve data collection, analysis, closing the gap between “what we want to be” in a given context and “where we are now,” formulating creative action paths to achieve results, and determining the key, underlying aspects of a role(s) (in the case of Organization Diagnosis, a whole organization) and the most salient and important strengths therein.