Rauch's comparison local result, while Toponogov's comparison is global.

For the upper curvature bound an analog of Toponogov's comparison holds only locally and indeed it follows from Rauch's comparison. There is a gloabal version for upper comparison. For the curvature bound $\kappa\le 0$ it has an addition assumption that space is simply connected. If $\kappa=1$ then one has to assume that any closed curve shorter than $2\cdot\pi$ can be contracted in the class of closed curves shorter than $2\cdot\pi$. (The case $\kappa>0$ can be reduced to $\kappa=1$ by rescaling.)

Rauch's comparison local result, while Toponogov's comparison is global.

For the upper curvature bound an analog of Toponogov's comparison holds only locally and indeed it follows from Rauch's comparison. There is a gloabal version for upper comparison --- the so-called (generalized) Hadamard--Cartan theorem [9.65+9.67 in our book]: For the curvature bound $\kappa\le 0$ it has an addition assumption that space is simply connected. If $\kappa=1$ then one has to assume that any closed curve shorter than $2\cdot\pi$ can be contracted in the class of closed curves shorter than $2\cdot\pi$. (The case $\kappa>0$ can be reduced to $\kappa=1$ by rescaling.)

Rauch's comparison local result, while Toponogov's comparison is global.

For the upper curvature bound an analog of Toponogov's comparison holds only locally and indeed it follows from Rauch's comparison. There is a gloabal version for upper comparison. For the curvature bound $\kappa\le 0$ it has an addition assumption that space is simply connected. If $\kappa>0$ then one has to assume that any closed curve shorter than $2\cdot\pi$ can be contracted in the class of closed curves shorter than $2\cdot\pi$.

Rauch's comparison local result, while Toponogov's comparison is global.

For the upper curvature bound an analog of Toponogov's comparison holds only locally and indeed it follows from Rauch's comparison. There is a gloabal version for upper comparison. For the curvature bound $\kappa\le 0$ it has an addition assumption that space is simply connected. If $\kappa=1$ then one has to assume that any closed curve shorter than $2\cdot\pi$ can be contracted in the class of closed curves shorter than $2\cdot\pi$. (The case $\kappa>0$ can be reduced to $\kappa=1$ by rescaling.)