My take on the differences in Martin's model:1. What Halliday called "context of situation" was renamed as "Register" in the Martin approach. In the Halliday model, a register is "a variety of language, corresponding to a variety of situation", so is basically the language pattern appropriate to a context of situation. Martin lifted the term up to apply to the situation, not to the language used in the situation.2. Martin and others (Rothery, Christie, etc.) added in a level of Genre above context of situation. As Annabelle said, this change was never adopted by Halliday, Hasan, and their followers. But it has proved popular in the area of Language Education, where it has proved useful to have individual genres existing as names in the theory, while in Halliday's approach, what a genre exists as a set of contextual features scattered across Field, Tenor and Mode (except since Matthiessen 2015 where terms like expounding, reporting, recreating, etc. are covered under Field).3. Halliday's "Semantic" stratum is largely (but not entirely) a level of meaningful abstraction over the clause (or clause complex). Martin's "Discourse Semantics" has been described as "grammar above the clause", e.g., capturing patterns which don't respect the borders of clauses, e.g., reference, evaluation, logical relations, exchange, etc. Neither characterisation is totally true, but there is definitely a difference in orientation here. And certainly Hasan's work explored deeply patterns outside the clause (she introduced the notion of cohesion, cohesive harmony, schematic structure etc.)4. Semantics was not highly specified by Halliday himself, and those who followed have proposed different components to the semantics. Most share some kind of Experiential and Logical semantics, and Speech Function as part of an Exchange semantics, some form of Thematic Progression, and Cohesion resources. The nature of these descriptions varies though, Hasan's "message semantics" takes a different turn that Eggin's (in the Martin camp) speech function network. And approaches to cohesive resources, while a common base in Hasan's work, varies in the way it is applied. Martin's model adds in other areas of semantics, e.g., Martin and White added in attitudinal, engagement and graduation models (part of Appraisal). And other components as well.
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[1] To be clear, Martin (1992) rebranded Halliday's context — from potential to instance — as register. Context of situation is merely an instance of context. Despite modelling register as context instead of language, an instance of Martin's register is nevertheless an instance of language: a text. This inconsistency is exacerbated by Martin's acknowledgement that instantiation does not cross stratal boundaries (such as from context to the strata of language.
[2] To be clear, in Halliday's model, register is a subpotential of language, a point of variation at the midway point on the cline of instantiation between potential and instance. As such, it realises context at a point of variation at the midway point on the cline of instantiation between potential and instance: a situation type. Because situation is an instance of context, it is realised by an instance of language: text, not register.
[3] To be clear, this is a serious misunderstanding of the stratification of levels of symbolic abstraction. Locating genres, as text types, above context is misconstruing varieties of language as not being language and as more abstract than culture.
[4] To be clear, Martin's notion of genre is unwittingly scattered over various dimensions of SFL theory. As text type, genre is register viewed from the instance pole of the cline of instantiation. The types of genre are categories within rhetorical mode, context, and the stages of genres are semantic structures of text types.
[5] To be clear, Halliday & Matthiessen (1999: 604) outline the explanatory advantages that motivate modelling semantics at a low level of abstraction above the grammar:
But in modelling the semantic system we face a choice: namely, how far "above" the grammar we should try to push it. Since the decision has to be made with reference to the grammar, this is equivalent to asking how abstract the theoretical constructs are going to be. We have chosen to locate ourselves at a low point on the scale of abstraction, keeping the semantics and the grammar always within hailing distance. There were various reasons for this.
First, we wanted to show the grammar at work in construing experience; since we are proposing this as an alternative to cognitive theories, with an "ideation base" rather than a "knowledge base", we need to posit categories such that their construal in the lexicogrammar is explicit.
Secondly, we wanted to present the grammar as "natural", not arbitrary; this is an essential aspect of the evolution of language from a primary semiotic such as that of human infants.
Thirdly, we wanted to explain the vast expansion of the meaning potential that takes place through grammatical metaphor; this depends on the initial congruence between grammatical and semantic categories.But in any case, it is not really possible to produce a more abstract model of semantics until the less abstract model has been developed first. One has to be able to renew connection with the grammar.
[6] To be clear, "patterns which don't respect the borders of clauses" are modelled in terms of cohesion in SFL Theory (Halliday & Hasan 1976). Martin (1992) rebranded these lexicogrammatical systems as his discourse semantic systems, rebranding reference and ellipsis-&-substitution as identification, lexical cohesion as ideation, and cohesive conjunction as conjunction (later connexion).
[7] To be clear, reference, evaluation, logical relations and exchange are all Halliday, not Martin. But see [10] below.
[8] This is misleading because it is untrue. See, for example, the 618-pages of Halliday & Matthiessen (1999).
[9] This is potentially misleading, since speech function is Halliday, not Martin.
[10] The systems of APPRAISAL differ from all the other discourse semantic systems in that they were developed through the collective efforts of Martin's colleagues and PhD students, with a first, early version of the key subsystem of ATTITUDE set out in 1994 by Iedema, Feez and White in their monograph, Media Literacy. Martin is possibly often regarded as the founder of the theory on the basis of references he made to APPRAISAL in a 1997 paper ("Analysing Genre: Functional Parameters") and/or his fuller treatment in his 2000 paper ""Beyond Exchange: Appraisal Systems in English". (The 1997 chapter presented an account of the sub-system of APPRECIATION, which was also developed through collaboration, most notably through Rothery's work on visual arts education.) However, key elements of what is now the widely referenced version of the theory were first presented in White's 1998 PhD thesis and then in White's 2002 paper, "The Language of Evaluation and Stance" (Handbook of Pragmatics.)". The version of the theory now most widely deployed in textual analysis was outlined in Martin and White's 2005 The Language of Evaluation: Appraisal in English. It includes accounts of the following sub-systems: AFFECT (drawing on earlier work by Martin), JUDGEMENT (collaboration by Martin, Iedema, Feez and White), APPRECIATION (collaboration by Martin and Rothery), ENGAGEMENT (from White's 1998 PhD thesis), GRADUATION (White and Martin collaboration).
