Full Idea
Dedekind proves mathematical induction, while Peano regards it as an axiom, ...and Peano's method has the advantage of simplicity, and a clearer separation between the particular and the general propositions of arithmetic.
Gist of Idea
Induction is proved in Dedekind, an axiom in Peano; the latter seems simpler and clearer
Source
report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Bertrand Russell - The Principles of Mathematics §241
Book Reference
Russell,Bertrand: 'Principles of Mathematics' [Routledge 1992], p.248