(Sorry for bad English.)
Let $f:\mathbb R^n\to\mathbb R^m$, $x\in\mathbb R^n$. What is a drivative $f'(x)$? It is the linear map $f'(x):\mathbb R^n\to\mathbb R^m$, $h\mapsto (f'(x))(h)=:f'(x)h$.
But what is a differential $df$? Usually people say $df(x)=f'(x)h$, but what is $h$? I tried to write it more correctly: $df(x,h)=f'(x)h$. I compared it with above. I see that simply $df=f'$. I'm confused a bit.