W-types vs Inductive types
(I'm assuming that by 'axiom schemas', you have in mind the work of Gimenez)

Extensionally, W-types and Gimenez' axiom schemas are equivalent. 

However, in an intensional setting, you won't go far with W-types: they are too extensional (by the very definition of the encoding) to be fit for programming. This has been discussed by several authors, especially :


Conor McBride: http://mazzo.li/epilogue/index.html%3Fp=324.html
Peter Dybjer, "Representing inductively defined sets by wellorderings in Martin-Löf's type theory"
Guogen &amp; Luo, "Inductive data types: well-ordering types revisited"