Your question is very general, but most relevant. It is indeed essential to be aware of these differences.
The general tendency is a move away from language as structure towards language use with references to Wittgenstein (meaning as use) and speech act theory (speaking as acting; how to do things with words).
A first important publication was Foucaults "L'ordre du discours" (introductiry lecture at the Collège de France in 1970). Foucault's poststructuralist and postmodernist theory links discourse to implicit power structures.
The power perspective is not general in discourse analysis. Researchers coming from different disciplines take different angles. Sociologists put the emphasis on sense construction, whereas linguists stress the pragmatic aspect by opposing text and discourse. In both cases a reconstruction of social reality via text analysis is an important point. Norman Fairclogh "Discourse and Social Change" (1992) provides a practical overview.
The discourse concept of Habermas and Apel is different from the fore mentioned line. There focus is on communicative acting.
There are certainly different perspectives guiding discourse analysis since language is an influential factor dominating human emotions and actions. Clearly, the text is only the surface manifestation of discourse- it is the tip of the iceberg. Discourse analysis helps to unravel and fathom out macro-elements that stand quite aloof from the text itself.As such, discourse analysis may be approached from two different perspectivizations: from within and from without. The former involves a pragmalinguistic analysis whereas the latter entails a sociopragmatic investigation. Pragmalinguistic analysis focuses on the features of the text or texture. Language system structural elements related to the context of situation or register may be used as a point of reference to analyze a given text (e.g, metadiscourse markers, hedges, nominalized structures, etc.) By contrast, sociopragmatic analysis concentrates on sociocultural and socioideological factors hidden in the text and indirectly signaled by lexicogrammer mechanisms.
Wish you good luck in the wonderful world of discourse analysis.
I also suggest that you read the preface of Genre in the Classroom: Multiple Perspectives, edited by Ann Johns, which shows approaches to Genre Analysis and Discourse analysis.
, ‘Discourse’ to a conversational analyst is basically social interaction; hence discourse analysis (DA) is the study of what speakers do and how they do it, how speakers take turns, and how this reveals speakers roles, their patterns of individual relationships, and their positions within larger institutional structures (Trask, 1999). To those doing interactional sociolinguistics, discourse is a means of studying how culture influences or controls interactional moves. Hence, DA analyzes how socio-cultural meanings and practices are played out in discourse. To those who view discourse as the medium for the expression of ideology, power and domination (i.e. critical discourse analysts), discourse is defined in terms of preferential access to and control over public discourse by social groups or organizations. Here DA focuses on the study of domination or power abuse based on ideology, gender, class, ethnicity, or sexuality (van Dijk, 2006). To those who view discourse as social semiosis, discourse goes beyond the use of human languages to include visual images, gestures, proxemics, symbols, sounds, colours and other forms of multimodal discourse. Thus, DA investigates how the various semiotic activities communicate meanings in the context of people and situations. These and other different assumptions and approaches to DA acknowledge that discourse must be contextually situated reflecting social situations of people’s everyday lives and experiences among others (See Van Dijk's Discourse studies: a multidisciplinary introduction).
Sometimes I work with Legal discourse which intends to make everybody equal in the face of the law and within the idea of “transparency” of language and which naturalizes the inequalities within social formations and put apart subjects with low literacy degree. The theorie of Discourse Analysis (PÊCHEUX, 1988) brings into focus the social, ideological and psychological path of the discourse, as well as the ritualized means of hegemonic subjugation. In this theory, the subject of discourse is an effect of language, and there is not only one possible meaning as legal discourse intends to show.