Why does this mismatch between skeleton and mesh occur? If I have align the mesh to the skeleton in Object Mode, when I switch to Edit Mode, the body falls to the ground, and vice versa.
It could be that you didn't applied the transformations. In object mode select the body and the armature (⇧ Shift + Left Click ) and then Ctrl + A and click on Apply all Transforms.
The problem was that the armature was in Pose Mode, so the pose was in the correct position, but the rest position was the one that appeared when you enter in Edit Mode.
To be sure, go to the right menu and click the button that looks like a stickman and click on Rest position.
After that, go back to object mode and select everything (body and armature) after that, press R, X and
-90 to rotate everything to the "normal" position, after that press Ctrl + A and apply all transforms.
Then, click the armature and press Tab to enter to Edit Mode and select only the head of the big bone below the body.
Press R, Z and
-90 so the bone will be looking to the front
With that the model will be at the right position in Object Mode, Edit Mode and Pose Mode.
Remember to change from Rest Mode to Pose Mode in the stickman menu when you want to change the pose of the character.