You are correct; exploratory factor analysis (EFA) and confirmatory (CFA) may be considered "factor analysis methods." Though strictly speaking, as @ttphns pointed out, EFA is technically factor analysis, and CFA is factor analysis constrained to a factor structure of substantive interest.
While EFA can be used to explore the factor structure of the data, it does not automatically give you an answer as to what structure you should test with a CFA. To do this, you must consider item content (i.e., what do the items say?) as well as substantive theory (i.e., what kind of factor structures does data similar to yours look like in the literature). Additionally, you do not necessarily have to run an EFA prior to a CFA. For example, I often work with language proficiency assessments, where items are written explicitly to measure listening, reading, writing, or speaking. Given this, I seldom run EFA before CFA, given I already have a good idea of the factor structure. I still run CFA, however, as testing whether the model fits the data is still important.
This is all to say that EFA does not tell you the correct structure for your research purpose; it is only a tool to explore the structure of your data and cannot be used a test a substantively relevant hypothesis regarding latent factors. In your question, you ask, "What extra benefit do I gain by testing my hypothesis using CFA?". While a good question, it does not make much sense given EFA is seldom ever one's final step when testing a hypothesis concerning latent variables. Your question should instead be, "In your question, you ask, "What extra benefit do I gain by using EFA?" because, in some instances (i.e., when you already have a plausible hypothesis regarding the factor structure), it may not be necessary.
Personally, I find reading the substantive journals that apply the methods I am interested in learning about (i.e., journals that apply factor analysis methods, as opposed to journals that discuss the more technical aspects of factor analysis) to be a great way to understand the rationale behind why certain methods are applied. In the references below, I have included a couple of articles in the patent-reported outcomes (PRO) literature that uses both EFA and CFA together.
DeWalt, D. A., Thissen, D., Stucky, B. D., Langer, M. M., Morgan DeWitt, E., Irwin, D. E., ... & Varni, J. W. (2013). PROMIS Pediatric Peer Relationships Scale: development of a peer relationships item bank as part of social health measurement. Health Psychology, 32(10), 1093.
Lai, J. S., Stucky, B. D., Thissen, D., Varni, J. W., DeWitt, E. M., Irwin, D. E., ... & DeWalt, D. A. (2013). Development and psychometric properties of the PROMIS® pediatric fatigue item banks. Quality of Life Research, 22(9), 2417-2427.