On when to use coordinates and other concrete constructions in mathematics, and when to use coordinate-free formulations and abstractions:

1. If your priority is to perform computations in mathematics, use coordinates and concrete constructions.
2. If your priority is to generalize to as broad a range of use cases as possible, use coordinate-free formulations and abstractions.
3. If your priority is to actually understand what is going on behind the mathematical formalism, learn how the coordinate-based and coordinate-free approaches are equivalent.

EDIT: in order to illustrate this point, let me give the concrete example that motivated this abstract statement. In my analysis class today I gave two proofs of Plancherel's theorem on compact abelian groups. The first approach used the Stone-Weierstrass theorem to reduce to showing that multiplicative characters separated points, and then for groups written explicitly in coordinates such as products of cyclic groups, or tori, one explicitly wrote down enough characters to separate points. This is a quick proof but relies on coordinates. Then I gave the coordinate free proof in which one simultaneously diagonalized convolution operators to obtain an eigenbasis of characters. This is more advanced (it requires the spectral theorem for compact operators) but has the advantage of generalization, for instance to the Peter-Weyl theorem for compact nonabelian groups.

Though even the concrete proof involved one abstraction step: in order for the Stone-Weierstrass theorem to be useful, it had to be abstracted from the more concrete Weierstrass approximation theorem, which only applied to polynomials, and not to trigonometric polynomials.