In Milnor's Topology from the differentiable viewpoint, page 3, he said:
One thinks of the nonhomogeneous linear mapping from the tangent hyperplane at $x$ to the tangent hyperplane at $y$ which best approximates $f$. Translating both hyperplanes to the origin, one obtains $df_x$.
What is nonhomogeneous linear mapping? As far as I know, linear maps are all homogeneous.
A reasonable interpretation of a linear map in this context is the one of an affine map. A homogeneous affine map would be linear in the conventional sense. This is similar to the terminology one uses for systems of linear equations $Ax+b=0$. The left hand side is an affine map and the system is called homogeneous if $b=0$. Also in calculus we frequently call a map $ax+b$ linear even though, strictly speaking, this map is affine.