In my opinion, methods are not about being accurate, but are more of whether they will best serve our purpose of the study. I would prefer to choose the method or methods that would enable me to answer my research questions. As to discourse and rhetorical analyses, my choice would depend on what I would like to know about persuasive language.
Rhetorical analysis focuses primarily on how language is intentionally used to achieve persuasive goals. A rhetorical analysis, for example, may describe how emotional appeals are used to persuade, or why some persuasive strategies are chosen over others, and how effective those strategies are in persuasion, and why.
Discourse analysis focuses primarily on how inferences may be made from linguistic elements in persuasive texts regarding social orders as well as persuaders' belief system and worldview, for instance. As Dr. John R. Yamamoto-Wilson concluded in a forum:
"I'd say that, properly approached, discourse analysis can open up a window of understanding about what individuals, organizations and even whole societies really think that cannot be accessed by simply listening to what they tell you they think."
https://www.quora.com/Why-do-discourse-analysis
However, some scholars see rhetorical analysis as part of discourse analysis. Literature I've read so far has shown no clear-cut difference between the two.
Hope my information is of any help in your deciding what method is likely to help you to best answer your research questions..
Depends if you are looking at the persuasion as a linguistic composition or a political remark.
In other words, the use of linguistic elements is the concern of the discourse-based analysis whereas the type of speech and the positioning of figure of speech is the rhetorical aspects of a persuasive speech.
Of course the boundaries become blurred with the focus of the research changing in the course of discussion.
These are sweeping statements as we may include visuals in persuasion where both are captured in linguistic and rhetorical analysis via different viewpoints.
Visuals may be broken down into visual grammar in linguistic analysis whereas visual may be discussed with cultural references and semiotic significations.
I am of the similar opinion with the others that the answer to the questions above very much depends on the objectives of the study and research questions. Discourse analysis is concerned more with formal aspects of language, such as syntactic structure, vocabulary, transitivity, and modality. For the analysis of such language resources, the works of Norman Fairclough (e.g. 1992; 2003; 2014) are excellent references.
The rhetorical analysis usually focuses on means of persuasion, that is, logos (knowledge, logical aspect), pathos, and ethos. In the modern rhetoric, this classical means of persuasion has been developed by a group of argumentation experts at the University of Amsterdam; they call their approach prama-dialectic (a dialogue between linguistic pragmatics and philosophy). They focus on strategic maneuverings that are manifest in three ways: topical selection, adaptation to audience demand, and rhetorical repertoire (later they call it lexical choices). Some of the references could be van Eemeren & Grootendorst (2004) and van Eemeren (2010).
If you have a spare time, my PhD thesis (2014) at the University of Malaya contains a number of sections and references on strategic maneuvering. If you live in Kuala Lumpur and are interested in the approach, you can see Dr. Teoh Mei Lin, head of English Department, Faculty of Languages and Linguistics at the University of Malaya. We were at Amsterdam from 2004 to 2006. I believe she must have a strong background in rhetorical analysis, particularly of the Amsterdam approach.
I agree wholeheartedly with much of what has been written in reply to your very good question. Recently, I have suggested (in writing about Donald Trump) that there are productive affinities between Aristotle's approach to rhetoric (ethos, logos and pathos) - which underlies much work in rhetorical analysis - and the work of Habermas on validity claims, rooted for him in 'universal pragmatics' with strong debts of speech act theory. Here is a link to the paper: https://www.researchgate.net/publication/309665229_Authenticity_Populism_and_the_Discourse_of_the_US_Presidential_Election
A much extended and updated version of this paper should be available soon as a contribution to the Journal of Language and Politics.
For an application of Habermas on validity claims please see: https://www.researchgate.net/publication/249731814_Speaking_sincerely_Public_reactions_to_the_death_of_Diana
There is also a section on validity claims in paul simpson's book "On the Discourse of Satire" (2003)
Working Paper Authenticity, Populism and the Discourse of the U.S. Preside...
Article Speaking sincerely: Public reactions to the death of Diana
It all depends on the objective of the research you are following. Notably, rhetorical analysis is concerned with the HOW of persuasion- the strategies writers use to convince a particular audience about a specific illocution they have in their heads. By contrast,discourse analysts seek to analyze language in terms of textual and contextual features which dominate and hide the intention which the writer is trying to evoke. As a consequence, discourse analysis comprises both the HOW and WHAT of the created discourse.On this basis, neither approach is more accurate than the other; the use of one rather than the other is simply related to the researcher's priorities.
In Gold mining (first chapter on Research gate), I discuss the relarion between critical rhetorical analysis and forms of discourse analysis.
I try to show how the rhetorical dynamics of narratives, comparisons, argumentation as well as contextuale framing relate to formal aspects of the discourse world that the rhetor constructs.
If you want to get access to the complete e-version of the book, please mail me.