The difference is
Routing grids : These are the grids which are used by place and route tool to route the METAL layers. Place and route tools like MAGMA/silicon ensemble or VDIO does this because this will automatically satisfy the DRC rule.
Manufacturing grid : Its nothing but related to process. Your layout should on on manufacturing grid else it will cause a problem in process. This is because while chip fabrication there is limit because of aperture angle (while masking) hence all your chip dimension has to satisfy this.