No, it’s not true in general (EGA 2, (1.2.3)).

The following example is taken from EGA 2, (1.3.3). Over a field $K$, let $S$ be the affine plane with a doubled origin. Then $S$ is the union of two affine open subsets $Y_1$ and $Y_2$, each of them is isomorphic to the affine plane, glued along the complementary subset of their origin. In particular, $Y_1$ is affine, but the open immersion $jcolon Y_1to S$ is not an affine morphism because the inverse image of $Y_2$ is $Y_1cap Y_2$,

the plane minus the origin, which is not an affine scheme.

On the other hand, if $S$ is separated, then the intersection of two open affine subschemes is affine, and this will imply that your desired result holds true (see EGA 1 (5.5.10)).

Finally, the other direction does not work either: every scheme, be it affine or not, is affine over itself.

What is true is that there is an antiequivalence between the category of schemes affine over $S$ (that is $S$-schemes for which the preimage of an open affine of $S$ is an open affine) and quasi-coherent $mathcal{O}_S$-algebras.

The anti-equivalence is realized by the pushforward of the structure sheaf and the relative spectrum (see Exercise 5.17 in Hartshorne’s *“Algebraic Geometry”*). This is of course not the result you hoped for, but maybe it is still interesting.