I cannot answer the question in a single post, but perhaps I can start.
Borrowing from Tom Scripps class, "DOE" held at the ASQ Quality Institute on Six Sigma, and my attendance in Shainin's Statistical Engineering class...
The purpose of conducting experiments to gain knowledge. I would characterize the differences in Fisher, Taguchi, Shainin as the depth of this knowledge gained by each design. Typically, you would start with a shallow, screening design and progress to focused, deep designs.
Fisher is the father of DOE. His work is referred to as Western or Classical school of DOE. The classical school first asserted Full Factorial designs (all variables and interactions are studied).
Taguchi works off the principle that most variation is attributed to only a few input variables. Study of all variables is not necessary. The Taguchi school asserts Fractional Factorial designs (study only a few variables).
Shainin's work is proprietary. I signed away my first born that I would not reveal his protected work. However, my opinion is that this is the most simplistic of the three. The designs are much smaller and quicker, but you give up the thoroughness of Fisher designs.
Each has strengths and limitations. Each has its own purpose (confirmatory, exploratory). It is very likely that, in the process of understanding your process, you will use more than one design.
-Steve.
