Given a $2$-dimensional surface $M \subset \mathbb R^3$ with a parametrization $(u,v) \mapsto X(u,v)$, one can define the gradient of a differentiable function $f$ on $M$ as
$$\nabla_Mf=\sum_{i,j=1}^2 g^{ij} \frac{\partial f}{\partial X_j}X_i,$$ where $X_1=\frac{\partial X}{\partial u}, X_2=\frac{\partial X}{\partial v}$ and \begin{equation} (g^{ij})_{i,j=1,2}:=\frac{1}{EG-F^2}\begin{pmatrix} G & -F \\ -F & E \end{pmatrix}. \end{equation}
But also, in vectorial notation, we have the surface gradient of $f$ defined as the Euclidean gradient minus the normal proportion, e.g.
$\nabla_Mf=\nabla f- nn^T\nabla f.$
I get both formulas, but what is the advantage/disadvantage over one another? In which cases is one notation more useful/practical over the other?
In my opinion, as long as you are not dealing with some nasty calculations, like dot products for normal fluxes, contracted on calculations etc you are safe with both.
It really depends on your personal taste. I started with vectorial notations then graduated to index notations and eventually worked fluently in both. One thing though: once I was working with non-isotropic things and working with vectorial notations adds more hassle than the problems it solves.
I still use both nowadays. Some theorems are stated a bit nicer in vectorial forms. Some calculations are made more explicit and accurate in index notations.
Some food for thought: Windows or Mac? Or Linux? Perhaps not a clear answer: I game on Windows and work on Mac. I run my servers using Linux. It depends on what I do. Hopefully this helps.