Suppose we only care about operads in chain complexes, although i think this can all work more generally. Then an operad is a monoid in the category of symmetric sequences with respect to a particular monoidal product. so a morphism of operads would then be a morphism of monoids. I think this framework might help clarify things. The reference I have in mind, although i am sure there are earlier ones, is Kathryn Hess's lecture notes on the cobar construction. You want to look at the second lecture, page 9 specifically.

This different monoidal product is just what you want it to be in order for a monoid to be an operad! I would explain more, but I can't do any better than Hess:
http://sma.epfl.ch/~hessbell/Minicourse_Louvain_Notes.pdf

PS: a symmetric sequence is a functor from the groupoid $\Sigma$ (where the objects are sets {1}, {1,2}, ... {1,...,n},... and the morphisms are bijections) to chain complexes.