When you switching between child theme and parent theme your widgets and theme options could be different because the child theme does not inherit all theme options of the parent theme and the parent theme does not inherit theme options of the child theme.
If you activate the child theme and settings that come from Theme Options look differently, activate the parent theme back > export theme options (Theme Options > Export/Import) and then import them after switching to the child theme. Widgets should be moved to correct areas at Appearance > Widgets.
