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Hi, Lorenzo,

The difference you ask about is quite involved with many nuances; many books & papers have been written about it. In a nutshell, though, the key difference is that

1. the N-P hypothesis test controls error rates (deductively, in the sense that the rates assume simple (point, rather than composite) null & alternative hypotheses, Ho & H1, respectively), &

2. the Fisherian significance test aims to provide evidence against Ho.

These 2 goals might seems similar, but in fact, the way they are worked out, the goals are almost always mutually exclusive.

Yes, a key difference is the emphasis on power for explicit alternatives in the Neyman-Pearson approach.

Here is an article reviewing the differences: 

Goodman SN (1993). P-values, hypothesis tests and likelihood: Implications for epidemiology of a neglected historical debate. Am J Epidemiol 137: 485-496.

Coverage of the differences can also be found in chapter 10 of Modern Epidemiology 3rd ed. (Rothman et al. 2008).

One reference that may help you understand the difference between the Neyman-Pearson approach and the Fisher approach this article by Neyman:

"Inductive Behavior" as a Basic Concept of Philosophy of Science, Revue de l'Institut International de Statistique / Review of the International Statistical Institute, Vol. 25, No. 1/3 (1957), pp. 7-22.

It's available online at http://www.jstor.org/stable/1401671.

You can also find Neyman's article at:

http://drsmorey.org/bibtex/upload/Neyman:1957.pdf

  for free.

Dear Lorenzo,

Another useful paper is Christensen's 2005 paper in The American Statistician:

Christensen, R. (2005) Testing Fisher, Neyman, Pearson and Bayes. The American Statistician, 59(2), 121-126.

Others may correct me, but I don't think Fisher's significance test envisions a specific H1.