I can't go in on mathematics but more practical.
A PCA is dimension reduction based purely on data. That is there are no theoretical limitations on how to group the factors.
A CFA on the other hand have those restrictions. In extremely (!) broad terms, you can see the CFA as a more complex Cronbach's alpha.
So I would say if you are on completely new waters a PCA/EFA would be good. If there are some literature that you can rely on go for CFA.
If you just want to reduce predictors for a regression you can do a PCA/PLS-regression.
Some onld notes that I have:
A central question for researchers developing instruments is whether to use exploratory (EFA) or confirmatory factor analysis (CFA).
Unfortunately, there are no generally accepted decision rules and there is continuing discussion about appropriate use of the two methods (Crowley and Fan, 1997; Hurley et al., 1997). Source Bates, R., Kauffeld, S., & Holton III, E. F. (2007).
An exploratory factor analysis (EFA) followed by a confirmatory factor analysis (CFA) were conducted for data analysis (Teddlie and Tashakkori, 2009; Gaskin, 2013c).
EFA is data driven whereas CFA is based on theory and/or empirical research (Suhr, 2006).
Factor analysis can be used as either an exploratory or confirmatory technique, depending on the final objective of research (Schervish, 1987).
EFA is normally the first step in building scales or a new metrics.Yong, A. G., & Pearce, S. (2013, p.79).
EFA should be followed by CFA using a different sample (or samples) to evaluate the EFA-informed a priori theory about the measure’s factor-structure and psychometric properties. Cabrera-Nguyen, P. (2010).
Regardless of how effectively the researcher believes item generation has reproduced the theorized latent variables, we believe that the initial validation of an instrument should involve empirically appraising the underlying factor structure (i.e., EFA). Source Roger L. Worthington and Tiffany A. Whittaker (2006, p. 815).
When developing new scales, researchers should conduct an EFA first, followed by CFA. Worthington, R. L., & Whittaker, T. A. (2006); Cabrera-Nguyen, P. (2010).
EFA is often considered to be more appropriate than CFA in the early stages of scale development because CFA does not show how well your items load on the non-hypothesized factors (Kelloway, 1995). Source Hurley, A. E., Scandura, T. A., Schriesheim, C. A., Brannick, M. T., Seers, A., Vandenberg, R. J., & Williams, L. J. (1997).
Research has shown that exploratory methods (i.e., principal-axis and maximum-likelihood factor analysis) are able to recover the correct factor model satisfactorily a majority of the time (Gerbing & Hamilton, 1996). Source Worthington, R. L., & Whittaker, T. A. (2006).
Researchers often erroneously assume that CFA is only used to verify or confirm hypothesized models, but researchers often apply CFA in an exploratory manner. Schmitt, T. A. (2011, p.315).
CFA is related to EFA, but is a theory-driven technique that tests the extent the proposed factor structure is replicated in another sample (Schreiber et al., 2006).
Most uses of ‘‘confirmatory’’ factor analyses are, in actuality, partly exploratory and partly confirmatory in that the resultant model is derived in part from theory and in part from a respecification based on the analysis of model fit.’ (Gerbing and Hamilton, 1996, p. 71).
Researchers often use MIs to modify CFA models, but when this occurs the perceived CFA becomes exploratory in nature (Bollen, 1989; Brown, 2001) and may be inappropriate.
Researchers should carefully consider all possibilities when a hypothesized model does not fit and realize that EFA is often more suitable for further “exploration” of poor fitting CFA models. Schmitt, T. A. (2011, p.315).
Please refer to Do Not Reify “Exploratory” versus “Confirmatory". Kline, R. B. (2015, p.197). Principles and practice of structural equation modeling. Guilford publications.