This Wikipedia article seems to suggest that the two concepts are somehow related but I don't see exactly how the pushforward of a vector field is a generalization of the differential of a map.
To clarify: if $\varphi: M \rightarrow N$ is a smooth map between smooth manifolds, the differential/pushforward of $\varphi$ at $x$ is a linear map $$d \varphi_x : T_x M \rightarrow T_{\varphi(x)} N$$ whose exact definition depends on the definition one uses for tangent vectors (but all of them are equivalent). If tangent vectors are defined as derivations acting on smooth real-valued functions, then the pushforward is given by
$$d \varphi_x (X)(f) = X (f \circ \varphi)$$
for an arbitrary function $ f\in C^{\infty }(N)$ and an arbitrary derivation $X\in T_{x}M$ at point $x\in M$. By definition, the pushforward of $X$ is in $ T_{\varphi (x)}N$ and therefore itself is a derivation, $ d\varphi _{x}(X):C^{\infty }(N)\to \mathbb {R} $.
On the other hand, each smooth vector field $ X:M\rightarrow TM$ on a manifold $M$ may be regarded as a differential operator acting on smooth functions $f(p)$ (where $p\in M$ and $f$ of class $C^{\infty }(M))$ when we define $X(f)$ to be another function whose value at a point $p$ is the directional derivative of $f$ at $p$ in the direction $X(p)$. In this way, each smooth vector field $X$ becomes a derivation on $C^{\infty}(M)$. Furthermore, any derivation on $C^{\infty}(M)$ arises from a unique smooth vector field $X$.
What I was thinking is, if we consider that the derivations in the above definition of the differential are vector fields, could it be considered a special case of the pushforward of a vector field (in the case in which it can be well defined as a vector field on $N$)? (I won’t reproduce the definition of pushforward here because it’s very long, but you can find it on the Wikipedia article…)