Methodology has to do with the strategies and heuristics through which we attempt to understand a complicated empirical reality (link). Our methodological assumptions guide us in the ways in which we attempt to collect data, the kinds of data we collect, the explanatory hypotheses we bring forward for that range of empirical findings, and the ways we seek to validate our findings. Methodology is to the philosophy of social science as historiography is to the philosophy of history.
Realism is also a set of assumptions that we bring to empirical investigation. But in this case the assumptions are about ontology -- how the world works, in the most general ways. Realism asserts that there are real underlying causes, structures, processes, and entities that give rise to the observations we make of the world, natural and social. And it postulates that it is scientifically appropriate to form theories and hypotheses about these underlying causes in order to arrive at explanations of what we observe.
This description of realism is couched in terms of a distinction between what is observable and what is unobservable but nonetheless real -- the "observation-theoretic" distinction. But of course the dividing line between the two categories shifts over time. What was once hypothetical becomes observable. Extra-solar planetary bodies, bosons, and viruses were once unobservable; they are now observable using various forms of scientific instrumentation and measurement. So the distinction is not fundamental; this was an essential part of the argument against positivist philosophy of science. And we might say the same about many social entities and structures as well. We understand "ideology" much better today than when Marx theorized about this idea in the mid-19th century, and using a variety of social research methods (public opinion surveys, World Values Survey, ethnographic observation, structured interviews) we can identify and track shifts in the ideology of a group over time. We can observe and track ideologies in a population. (We may now use a different vocabulary -- mentality, belief framework, political values.)
There are several realist methodologies that are possible in the social sciences. The methodology of paired comparisons is a common part of research strategies in the historical social sciences. This is often referred to as "small-N research." (Here is a description of the method as practiced by Sid Tarrow; link, link.) The method of paired comparisons is also based on realism and derives from causal ideas; but it is not specifically derived from the idea of causal mechanisms. Rather, it derives from the simpler notion that causal factors function as something like necessary and/or sufficient conditions for outcomes. So if we can find cases that differ in outcome and embody only a small number of potential contributing causal factors, we can use Mill's methods (or more general truth-table methods) to sort out the causal roles played by the factors. (Here is a discussion of some of these concepts; link.) These ideas contribute to methodology at two levels: they give the investigator a specific idea about how to lay out his/her research ("seek out relevantly similar cases with different outcomes"), and they embody a method of inference from findings to conclusions about causal relations (the truth-table method). These methods allow the researcher to arrive at statements about which factors play a role in the production of other factors. (This is a logically similar role to the use of multiple regression in quantitative studies.)
It is an interesting question to consider whether realism in ontology leads to important differences in methodology. In particular, does the idea that things happen as the result of an ensemble of real causal mechanisms that can be separately understood lead to important new ideas about methodology and inquiry?
Craver and Darden argue in In Search of Mechanisms: Discoveries across the Life Sciences that mechanisms theory does in fact contribute substantially to contemporary research in biology, at a full range of levels (link). They maintain that the key goal for much research in contemporary biology is to discover the mechanisms that produce an outcome, and that a central component of this methodology is the effort to explain a given phenomenon by trying to fit one or more known mechanisms to the observed process. So working with a toolbox of known mechanisms and "problem-solving" to account for the new phenomenon is an important heuristic in biology. This approach is both ontological and methodological; it presupposes that there are real underlying mechanisms, and it recommends to the researcher that he/she be well acquainted with the current inventory of known mechanisms that may be applied to new settings.
I think there is a strong counterpart to this idea in a lot of sociological research as well. There are well understood social mechanisms that sociologists, political scientists, and other researchers have documented -- easy riders, prisoners dilemmas, conditional altruism -- and the researcher often can systematically explore whether one or more of the known mechanisms is contributing to the complex social outcomes he or she is concerned with. A good example is found in Howard Kimeldorf's Reds or Rackets?: The Making of Radical and Conservative Unions on the Waterfront. Kimeldorf compares two detailed case histories and strives to identify the concrete social mechanisms that led to different outcomes in the two cases. The mechanisms are familiar from other sociological research; Kimeldorf's work serves to show how specific mechanisms were in play in the cases he considers.
This kind of work can be described as problem-solving heuristics based on application of a known inventory of mechanisms. It could also be described as a "normal science" process where small theories of known processes are redeployed to explain novel outcomes. As Kuhn maintains, normal science is incremental but creative and necessary in the progress of science.
A somewhat more open-ended kind of inquiry is aimed at discovery of novel mechanisms. McAdam, Tarrow and Tilly sometimes engage in this second kind of discovery in Dynamics of Contention -- for example, the mechanism of social disintegration (kl 3050). Another good example of discovery of mechanisms is Akerlof's exposition of the "market for lemons" (link), where he lays out the behavioral consequences of market behavior with asymmetric knowledge between buyer and seller.
So we might say that mechanisms theory gives rise to two different kinds of research methodology -- application of the known inventory to novel cases and search for novel mechanisms (based on theory or empirical research).
Causal-mechanisms theory also suggests a different approach to data gathering and a different mode of reasoning from both quantitative and comparative methods. The approach is the case-studies method: identify a small set of cases and gain enough knowledge about how they played out to be in a position to form hypotheses about the specific causal linkages that occurred (mechanisms).
This approach is less interested in finding high-level generalizations and more concerned about the discovery of the real inner workings of various phenomena. Causal mechanisms methodology can be applied to single cases (the Russian Revolution, the occurrence of the Great Leap Forward famine), without the claim to offering a general causal account of famines or revolutions. So causal mechanisms method (and ontology) pushes downward the focus of research, from the macro level to the more granular level.
The inference and validation component associated with CM looks like a combination of piecemeal verification (link) and formal modeling (link). The case-studies approach permits the researcher to probe the available evidence to validate specific hypotheses about the mechanisms that were present in the historical case. The researcher is also able to try to create a simulation of the social situation under study, confirm as much of the causal internal connectedness as possible from study of the case, and examine whether the model conforms in important respects to the observed outcomes. Agent-based models represent one such set of modeling techniques; but there are others.
So the methodological ideas associated with CM theory differ from both small-N and large-N research. The search for causal mechanisms is largely agnostic about high-level regularities -- either of things like revolutions or things like metals. It is an approach that encourages a more specific focus on this case or that small handful of cases, rather than a focus on finding general causal properties of high-level entities. And it is more open to and tolerant of the possibility of a degree of contingency and variation within a domain of phenomena. To postulate that civil disorders are affected by a group of well-understood social mechanisms does not imply that there are strong regularities across all civil disorders, or that these mechanisms work in exactly the same way in all circumstances. So the features of contingency and context dependence play an organic role within CM methodology and fit badly in paired-comparisons research and statistical modeling approaches.