Though I am late for this answer by almost 7 years, but still want to give a try. I have the following picture in mind.
$$\text{Cotangent Bundles}\leftrightarrow \text{Pull-backs}\leftrightarrow \text{Differentials}$$ $$\text{Tangent Bundles}\leftrightarrow \text{Push-forward}\leftrightarrow \text{Tangent Vectors}$$
Why the preference toward "co"?
One approach that comes to my mind is from Warner. Let $M$ be a manifold and $(U,\phi)$ and $(V,\psi)$ are two coordinate systems about $m$. If $\phi=(x_1,x_2,...,x_n)$ and $\psi=(y_1,y_2,...y_n)$, then we note that $$\frac{\partial}{\partial y_j}\Biggr|_m =\sum_{i=1}^n\frac{\partial x_i}{\partial y_j}\Biggr|_m \frac{\partial}{\partial x_i}\Biggr|_m $$
If we have $x_1=y_1$, then $\frac{\partial}{\partial y_1}\Bigr|_m \neq \frac{\partial}{\partial x_1}\Bigr|_m $ but $dy_1=dx_1$. This is because $\frac{\partial}{\partial x_i}\Bigr|_m $ depends on $\phi$ and not only on $x_1$. In this sense, differentials are more natural and hence cotangent bundles.