Hatcher contains the following paragraph:
Define a reparametrization of a path $f$ to be composition $f\psi$ where $\psi:I\to I$ is any continuous map such that $\psi(0)=0$ and $\psi(1)=1$. Reparametrizing a path preserves its homotopy class since $f\psi\simeq f$ via the homotopy $f\psi_t$, where $$\psi_t(s)=(1-t)\psi(s)+ts$$ so that $\psi_0=\psi$ and $\psi_1(s)=s$.
What does it mean to say that reparametrization preserves the homotopy class of a path? Aren't $f$ and $f\psi$ the same curves anyway? Why would one mention $f\psi\simeq f$?