Navy tugs need damage control more than grey paint

Is there any design difference between common tugs and navy tugs besides the grey paint?

Yes, compared with a basic tug, but a civilian tug designed for explosive freight or harbour firefighting may have equivalent protection.

Naval tugs usually have extra subdivision because they move warships loaded with explosives. They should also carry full harbour-firefighting equipment, although ordinary tugs often have some firefighting capability. A civilian tug already equipped for explosive freight or harbour firefighting may therefore have much the same provisions. The relevant distinction is the work and hazard the tug is designed for, so grey paint alone does not identify a different engineering requirement.