This is a question about notation. Of course, as long as the notation is clearly defined, it doesn't matter at all which notation we use, but it's still helpful to ask about a few possible confusions that can be caused by the usage of $\nabla$.
- Laplacian. Sometimes, the Laplacian is denote $\nabla^2,$ so $\nabla^2 u = \nabla \cdot (\nabla u).$ Although this seems to justify the use of $\nabla^2,$ the Laplacian is not the only thing that comes from the square of $\nabla$. Another would be the "Hessian" $\nabla (\nabla u).$ (Well, it is the Hessian up to index raising/lowering.)
- Directional derivative. The $\nabla$ operator comes with two steps: (1) find the derivative of $u$ as a linear form, (2) raise an index to make it a vector, not a covector. So, if we come across a directional derivative of a vector like $(\mathbf f \cdot \nabla) \mathbf g,$ although it is tempting to drop the bracket to write $\mathbf f \cdot \nabla \mathbf g,$ we run into the problem that $\nabla \mathbf g$ is the tensor product of two vectors, and we do not know with respect to which of the two vectors are we dotting $\mathbf f$ with if we write it without bracket. So this no-bracket notation fails.
Things get increasingly unnatural with many other common combinations of $\nabla$, and in a way, the notations used in differential topology (Lie derivative, exterior derivative, hessian as a 2-form, etc) are much clearer and more effective in avoiding such undesirable temptations described above.
The question I am going to ask is quite general: what are some good ways to avoid these possible confusions, and what recipie of notations or way of thinking would do a better job of avoiding these confusions? Although I do understand how these wired $\nabla$'s work, I personally feel uncomfortable working with notations that lead to such "ambiguity".