It might help to consider the extreme case when $x$ is a closed point of $X$, and $i$ is the inclusion $\{x\} \hookrightarrow X$. The pullback $i^{-1}\mathcal O_X$ is then the stalk of $\mathcal O_X$ at $x$, i.e. the local ring $A_{\mathfrak m}$, if Spec $A$ is an affine n.h. of $x$ in $X$, and $\mathfrak m$ is the maximal ideal in $A$ corresponding to the closed point $x$.
Now a single point, with a local ring $A_{\mathfrak m}$ as structure sheaf, is not a scheme (unless $A_{\mathfrak m}$ happens to be zero-dimensional).
Moreover, the restriction map from sections of $\mathcal O_X$ over $X$ to section of $i^{-1}\mathcal O_X$ over $x$ is not evaluation of functions at $x$ (which corresponds to reducing elements of $A$ modulo $\mathfrak m$), but is rather just passage to the germs of functions at $x$.
The idea in scheme theory is that sections of $\mathcal O_X$ should be functions, and restriction to a closed subscheme should be restriction of functions. In particular, restriction to a closed point should be evaluation of the function (if you like, the constant term of the Taylor series of the function), not passage to the germ (which is like remembering the whole Taylor series).
If you bear this intuition in mind, and think about the case of a closed point, you will soon convince yourself that the general notion of closed subscheme is the correct one: If we restrict functions to the locus cut out by an ideal sheaf $\mathcal I$, or (in the affine setting) by an ideal $I$ in $A$, then two sections will give the same function on this locus if they coincide mod $\mathcal I$ (or mod $I$ in the affine setting), and so it is natural to define the structure sheaf to then be $\mathcal O_X/\mathcal I$ (or to take its global sections to be $A/I$ in the affine settin), rather than $i^{-1}\mathcal O_X$.
