"Content analysis" is often used for a realist approach interested in how participants' talk reflect their ideas, thought, feelings, etc. "Thematic analysis" is used as an alternative term (preferred by many; see Braun and Clarke "Using thematic analysis in psychology"). In another research tradition, "content analysis" is an approach based on the quantification of words in text and talk. 

"Discourse analysis" is an umbrella term (see Wooffitt "Conversation Analysis and Discourse Analysis : a Comparative and Critical Introduction"). However, "discourse analysis" is often used for approaches interested in examining participants' linguistic repertoires and how these are linked to different representations of the social world, e.g. how different lexical choices bear on different depictions of disability, homosexuality, policital issues, etc. Unlike many types of content analysis, discourse analysis is agnostic about participants' 'real' thoughts, feelings as represented in their talk. Language is conceived as a vehicle for action (where the action being studied is the representation of reality; see Potter "Representing reality").

There is, then, "conversation analysis" which has a much broader focus on talk-in-interaction as a vehicle of social action.

Hope this helps!

Content analysis does not give you the possibility to pay scientific attention to the context, in which a verbal act is taken, and therefore limits the analysis of the verbal act. It actually does not even pay attention to verbalization as an act, that reacts and provokes – it rather takes it as a fact. With content analysis the intersubjective space is not involved that takes place in a interview. Rather, text is taken as a fact without the context of the discourse in which the text arises.

I will send a list of sources for discourse analysis e-books, all usually provide information about what you seek.

If you haven't read this paper, it's quite relevant to this question.  Schegloff, E. A. (1993). Reflections on quantification in the study of conversation. Research on language and social interaction, 26(1), 99-128.

As Marco says, content analysis can be either more qualitative or quantitative. P. Mayring has a good book on this issue, which is available here:

https://www.researchgate.net/publication/266859800_Qualitative_content_analysis_-_theoretical_foundation_basic_procedures_and_software_solution

I also think of discourse analysis as operating at two different levels. The replies you have received so far are mostly about the form which is similar to Conversation Analysis. The other is associated with Michel Foucault and is much more abstract.

Book Qualitative content analysis - theoretical foundation, basic...

As Dragica says, the most striking and important difference is in the role of context in the analysis. Nonetheless, I think that there is another important area, where they are different, namely the type of research questions and coding schemes they utilize:

Content analysis is working with more well- and pre-defined research questions then DA. This predefined focus determines which content will be searched and registered in the texts, thus, it also serves as a basis for the elaboration of the coding scheme. 

Discourse analysis (especially critical discourse analysis, and Wodak's discourse-historical approach) usually works with general research questions aiming to grasp implicit meanings and the very nature and characteristics of the social phenomenon analysed. Sometimes even the ‘codes’, in a certain sense, emerged in the course of the analysis, not stem from a pre-fixed coding scheme. The very nature of the analysis is that it allows the main focuses and topics emerge in the process.

If you are interested, I have a short methodological paper about the differences between (quantitative) content analysis and discourse analysis. If you are a subscriber of Sage Research Methods, then you can download it from this site: 

http://srmo.sagepub.com/view/methods-case-studies-2015/n13.xml?rskey=3pC7c7&row=4

If not, I can send it to you private. (Title: Content Analysis Versus Discourse Analysis: Examination of Corporate Social Responsibility in Companies' Homepage Texts)

Dear Zsuzsanna, is it possible to have a copy of your paper as well ?

One of the aims of Discourse Analysis is the attainment of CONSENSUS. (HABERMAS). Content analysis measures common meanings, a major differnce.

Qualitative content analysis is looking for patterns that prevail the content so as to generate a general conclusion or identify the norms and sometimes formulate a theory. In discourse analysis, the researchers, try to reveal how discourse serves other purposes, e. g., dominance, and how such purposes as power is exercised through discourse.

the difference is based on the definition of discourse. as i understand, in studies with discoursive approach, u should pay attention not only to the text but to another texts, context and even discourses. while, in studies based on qualitative content analysis, concerning the simple text and interperetion is sufficient to give analysis.

Discourse analysis necessarily involves 2 or more persons. That is not so for content analysis.

I found this text very helpful to distinguish between the two:

Herrera, Y. M., & Braumoeller, B. F. (2004). Symposium: Discourse and content analysis. Qualitative Methods, 2(1), 15-19.

I'm just writing a paper on how linguistic analysis can be used to assess and understand more deeply the verbal contributions of participants in qualitative research groups. One piece of work stands out in this regard: Russell Wilson's 2011 PhD thesis: 

Wilson, Russell. 2011. Appraisal Theory as a Linguistic Tool for the Analysis of Market Research Interview Data. Unpublished PhD Dissertation, Centre for Translation Studies, University of Leeds.

Not only does he use consumers' use of evaluative language to rank the strength and therefore importance of their statements about products and their liking for them, but he uses conversation analysis  and discourse analysis in tandem to rate how the position in the conversation (for example, a response or an initiation) of the utterance modifies its significance - for example, given what we know about politeness, If person A says 'I really like the shape' and person B responds 'Mmm, yeah', person B's utterance is not as strong an endorsement as person B's. Content analysis might have them down as 'both liking' the product. I'd give the thesis a look - it's available through the University of Leeds. 

Dear Zsuzsanna, is it possible to have a copy of your paper on the topic : Content Analysis Versus Discourse Analysis: Examination of Corporate Social Responsibility in Companies' Homepage Texts). 

content analysis may be qualitative and quantitative too but discourse is a qualitative method. content has nothing to do with the background of the issue but discourse is basically to explore the situation in the context. 

In content analysis you have clear and objective data with you  and you based on your research question dig into details and stick yourself to specific domain. in discourse analysis you think across the situation i.e, you do not only  focus on the context you are working but incorporate other overlapping contexts with due consideration of each context to dig deep.

regards

Qualitative content analysis is dividing into three types: conventional, directed and summative. each of them has their own implications but one of the most differential feature of qualitative than quantitative is pre-defined concepts by which you are going through the text. mostly, in qualitative you don't use any pre-assumed theoretical framework as a spotlight on the text words.

Discourse analysis involves multiple living participants. Content analysis can come from any textual source, i. e, books, articles, etc.

Thanks for naming the differences.

How do you think qualitative data analisis software can facilitate the dicourse analysis?

By the use of Evidenetials. Read my book Hermeneutics and Discourse Analysis in Social Psychology. Paper back Edition

Evidentials are factual assertions accompanied by the reasons for such assertions.

Difference between content and context