Tangent vectors to a manifold may be defined either by means of tangent curves or by derivations, as seen in this Wikipedia article.
The two books on manifolds I've been able to get my hands on (Lee's Introduction to Smooth Manifolds and Warner's Foundations of Differentiable Manifolds and Lie Groups) define tangent vectors by means of derivations. What book (or resource) develops the theory of tangent spaces, differentials, etc. by defining tangent vectors by means of curves?
Given that tangent curves are (I hope you can agree) more easily seen as a generalization of tangent vectors in $\mathbb{R}^n$, and that they allow us to define tangent vectors in $C^k$ manifolds (as opposed to derivations, which must be defined on $C^\infty$ manifolds), why is there a preference for using derivations as opposed to tangent curves?