What's the difference between time-dependent flow (isotopy) and time-independent flow?

I'll use Amitai Yuval's notation.

Notice that if $X_t=X$ does not dependent on $t$, then $\frac{d}{dt}\varphi_t(p)=X(\varphi_t(p))$ means that $\varphi$ is the flow of $X$. In particular, $\varphi_t\circ\varphi_s=\varphi_{t+s}$ for all $t,s\in\mathbb{R}$ (which is a defining property of the flow).

Conversely, if $\varphi_t\circ\varphi_s=\varphi_{t+s}$ $\forall t,s\in\mathbb{R}$, in particular $\varphi_t^{-1}=\varphi_{-t}$. By definition of $X_t$ we have: \begin{align*} X_t(p)&=\left.\frac{d}{ds}\right|_{s=t}\varphi_s(\varphi_t^{-1}(p))\\ &=\left.\frac{d}{ds}\right|_{s=t}\varphi_{s-t}(p)\\ &=\left.\frac{d}{du}\right|_{u=0}\varphi_u(p),\,\,\,\text{where }u=s-t. \end{align*}

This shows that $X_t$ does not depend on $t$. So we have just proved that:

$X_t$ depends on $t\Leftrightarrow$ $\varphi_t\circ\varphi_s=\varphi_{t+s}$ for all $t,s\in\mathbb{R}$.