I don't like using tensors. They don't provide any intuition when viewing the equations. With vector calculus, the grad, curl and divergence operators all provide some insight towards the physical process that the equation describes. Of course, if you're describing a 4-D space (general relativity) then you have no choice but tensors. Also a tensor is more compact for describing viscosity which is why it is sometimes used for writing the Navier Stokes equations (that's how they taught it at Stanford University). However I still think a vector calculus description is better for the Navier Stokes equations.

Gary