Researchers often use two measures that have very similar items and argue that they measure different things (e.g., "I always worry when I am around cars"; "I am fearful of cars"). Lets call the hypothetical measures the Fear of Cars Measure and Anxiety from Automobiles Scale. I am interested in testing empirically if they indeed assess different latent constructs, or if they measure the same thing.
The two best ways I can think to do this would be through exploratory factory analyses (EFA) or confirmatory factor analysis (CFA). I think EFA would be good because it allows all of the items to load freely without constraints. If items from the two scales load on the same factors, then I can conclude that the measures likely don't assess different things very well. I can also see the benefits in CFA, however, since I will be testing pre-defined models. For example, I could compare the fit of a model in which all items load onto a single factor (i.e., they don't assess different constructs) or the items are separated into the expected measures. An issue with CFA, I suppose, is that it would not really consider alternative models (e.g., a three factor model).
For the purposes of discussion, lets also perhaps consider that there may be two other very similar measures out there (e.g., Car anxiety questionnaire and the Scales for the assessment of car fears) that I wish to throw into the mix!
How can I best statistically determine if two measures assess different constructs?