If you look in your typical dictionary you will find that the word 'synergetics' does not appear there (at least, it didn't in mine), but the word 'synergy' does and is defined as "a combined effect of drugs, organs, etc., that exceeds the sum of their individual effects. It's derived from the Greek 'synergos' meaning working together. Synergetics was introduced in the late 1970s by Hermann Haken who noticed profound similarities between the fundamental mathematical models which are used to describe the cooperative behavior of various active systems in physics, chemistry, biology, and social sciences.
Synergetic is a science of structure and it seeks to find general laws that govern the formation of these structures. By structures one means everything from the shape of the galaxies to that of snow crystals; from ant colonies to biological cells; structures emerging out of chaos and of the human mind. The latter being, perhaps, most pertinent if we consider that physics, chemistry, psychology, sociology, theology etc., are all, in essence, structures of the human minds. If one is able to extract the governing dynamics of the formation of the structures of the human mind…we'll be able to see the universe and ourselves in a whole new way!
So, in a sense, synergetic can be thought of as a shop where fundamental models of cooperative behavior (structure formation) are worked up and their careful mathematical analysis is performed. The fact that it's possible to extract the laws and patterns of cooperative behavior of active systems seems, at first glance, to be surprising and not at all representative of biological or social systems. Even a single prokaryote represents an extremely complicated object, not to mention the extreme complexity of human beings who are individual elements of social systems. However! In cooperative interactions these elements often act is if they were simple units that can be described by a set of a few variables. Their vast internal complexity is not directly manifested in their interactions. This, perhaps, is a direct consequence of evolution for if one were to allow the elements to reflect all their internal complexity in the interactions, then the system as a whole would most probably not be able to display any stable and predictable behavior.
Thus, the wide variety found in the interactions and activities of the living systems emerges out of interplay of many simple ordered patterns that correspond to different structural levels. Ultimately, it is this notion that makes the mathematical description of the living systems feasible. (Note that concept of the order-parameter and the enslavement, which are at the heart of synergetics, emerged out of Ginzburg-Landau Theory of phase transitions).
It seems fair to conclude that this self-organizations phenomenon, or emergence of structures, differs in inorganic and biological systems not in the type of individual patterns but rather in the complexity of the emerging hierarchical structures with the latter system exhibiting higher complexity.