You are correct that in practice one does not care so much whether two functors are equal, but rather whether they are naturally isomorphic. The difficulty is that typically one does not want to completely forget about the natural isomorphism either! (Which is what happens if one quotients out by natural isomorphisms as you suggest.)
A typical example (which I hope will make sense to you) is to consider for each topological space the category of vector bundles on $X$. So we have a category $Vect\_X$. If $f: X \to Y$ is a continuous map of spaces,
and $\mathcal V$ is a vector bundle on $Y$, then one can pull-back $\mathcal V$ to form
a vector-bundle $f^* \mathcal V$ on $Y$. So we get a functor $f^*: Vect_Y \to Vect_X$.
If now $g: Y \to Z$ as well, then one sees that $(g f)^* $ is naturally isomorphic to
$f^* g^*$, say by some natural isomorphism $c_{f,g}: (g f)^* \cong f^* g^*.$
Morally, one would like to say that $X \mapsto Vect_X$ and $f \mapsto f^* $ gives a contravariant functor from the category of topological spaces to the category of categories Cat. In practice, because we don't have equality between $(f g)^* $ and $f^* g^* $, we don't get such a functor, although we do get a functor into your suggested category "Cat".
The problem is that in practice, one wants to remember the natural isomorphisms $c_{f , g}$,
which satisfy some important properties: for example, if $h: Z \to W$ is a third map,
then $f^* c_{g,h} \circ c_{f, h g} = c_{f,g}h^* \circ c_{gf, h}.$ (If you write this out, it is a commutative square that relates the various ways to pull back $\mathcal V$ by
$h$, $g$, and $f$, taking into account the associative law $(hg)f = h(g f)$.)
So really, one wants to work in a more sophisticated structure then either Cat or "Cat",
namely a structure in which the objects are categories, the morphisms are functors,
and in which we add an explicit extra layer of structure, so-called 2-morphisms, which are natural isomorphisms between functors. One then develops a theory in which one morally
regards functors as equal if they coincide up to natural isomorphism (i.e. up to a diagram involving a 2-morphism), but one also keeps track of the 2-morphisms.
The resulting structure is called a 2-category, and is part of the study of higher category theory. (Everything written above should provide some background for understanding Martin Brandenburg's answer.)
This theory has a lot in common with homotopy theory: in topology, one can pass to a category in which objects are spaces and maps are continuous maps modulo homotopy, but in lots of applications one wants to remember not just that maps are homotopic, but one actually wants to remember the homotopy; often then there are homotopies between homotopies,
homotopies between homotopies between homotopies, and so on. Similarly in category theory one can introduce not just the notion of 2-category, but notions of n-categories, in which
there are 3-morphisms beweeen the 2-morphisms, etc., up to n-morphisms.
Your structure "Cat" is analogous to passing to the homotopy category in topology; it is interesting, but forgets information one often wants to remember.
If you search for "higher category theory", you will find an enormous amount of material.
One good reference is the n-category cafe and the n-Lab. Another place to look is at the
various manuscripts on Jacob Lurie's web-page (at Harvard). What you will find is a rather intricate, and rapidbly evolving theory, blending category theory and homotopy theory in a fascinating (although sometimes daunting!) way.
In summary, your idea and your question are far from misguided, but in fact are pointing at one of the most active areas of modern research in category theory and related areas!