Difference between exploratory and confirmatory factor analysis in determining construct independence 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? 
 A: These methods are examples of application of exploratory and confirmatory data analysis. Exploratory data analysis looks for patterns while confirmatory data analysis does statistical hypothesis testing on proposed models.  It really should not be viewed in terms of which method to use it is more a matter of what stage in the data analysis you are at.  If you are unsure of what factors to include in your model you apply EFA.  Once you have eliminated some factors and settled on what to include in your model you do CFA to test the model formally to see if the chosen factors are significant.
A: If I understand your question correctly it is a question about testing. Then simply testing requires a kind of confirmatory factor-analysis, the same as the question: "do the means in the subgroups really differ?" requires a t-test.     
Unfortunately(?) with the selection of the general approach of the appropriate method of factor analysis also different mathematical (and statistical) models are often implied, for instance, if you select "CFA" in SPSS then it is implied that you assume uncorrelated errors and that uncorrelated errors are estimated and the estimation is excluded from the model - so, in my opinion, because of the further implications the initial selection of the correct factor analytic approach is often compromised by this mathematical/statistical implications.     
In short: your question is one of the sort "testing the null", thus you need CFA or better: the methods developed in the framework of SEM (structural equation modelling). Note, there is a friendly and helpful mailing list full of experts in SEM called "SEMNET" and since I'm not a real expert you might refine your feedback by asking there...
