There seems to be a general sentiment that triangulated categories are not the "correct" notion to use because mapping cones of morphisms are unique, but only up to non-unique isomorphism.
Does anyone know a concrete example of a "proof" or application that we would like to make with them, but which gets hindered because of this property?
Since I have already given a similar answer recently, I don't want to be branded as the "anti-triangular" guy: the formalism of triangulated categories can be useful in certain settings. That said they do have some definite shortcomings, and I would like to list a few in this answer. As a homotopy theorist, most of my examples will come from homotopy theory, but I'm sure there are others from algebraic geometry as well.
The first easy example is: there's no notion of algebra and module in a triangulated category such that for an algebra $A$ the category of $A$-modules is again triangulated. If you go looking at more refined approach you'll see that to define an associative algebra or a module over an associative algebra in a "well-behaved" way, you need to involve higher homotopies that you are forgetting when you are restricting yourself to the triangulated category.
This alone is a huge shortcoming: we like to be able to talk about categories of modules!
Another one, maybe more abstract, but important for computations is the following: There is no notion, as far as I know, of descent for triangular categories. That is, there's no way in which one could say "this family of triangular categories form a sheaf".
For example, let $B/A$ be a faithful Galois extension of commutative ring spectra with finite Galois group $G$, in the sense of Rognes' monograph. A good concrete example can be the $C_2$-extension $KO\to KU$. Suppose we know the Picard group of $B$ (for example because it is even periodic, or because it has some other property that makes it easy to compute). We would like to deduce some information about the Picard group of $A$.
Using a more refined approach (either ∞-categories or model categories or what have you), you can define an invariant of an algebra $C$ called the Picard space $\mathrm{Pic}\,C$, which is an $E_∞$-space such that $\pi_0\mathrm{Pic}\,C$ is the Picard group of $C$. Then you can prove that there's an equivalence $$\mathrm{Pic}\,A\cong(\mathrm{Pic}\,B)^{hG}$$ and obtain a spectral sequence relating the Picard group of $B$ to the one of $A$. In fact most of the actual computations of the Picard group of ring spectra I know are done in a similar fashion.
You could also do a more advanced version when your ring spectrum of interest is the global sections of some sheaf of ring spectra. This is how the Picard group of Tmf and TMF have been computed.
Another maybe more abstract way of exploiting descent is in putting symmetric monoidal structures on derived categories of schemes. If we denote the derived $\infty$-category of a scheme $\mathscr{D}(X)$, descent tells us that we can express it as a limit $$\mathscr{D}(X)\cong \lim_{\mathrm{Spec}\,R\subseteq X}\mathscr{D}(R)$$ where the limit is indexed by the affine open subsets. But now the diagram we are taking the limit of is a diagram of symmetric monoidal stable $\infty$-categories, so the limit inherits a symmetric monoidal structure. Concretely, the tensor product $P\otimes Q$ has the property that on every affine chart we have $$\Gamma(\mathrm{Spec}\,R ,P\otimes Q)\cong \Gamma(\mathrm{Spec}\,R,P)\otimes_R\Gamma(\mathrm{Spec}\,R,Q)$$ The traditional way of defining the tensor product on the triangulated category $h\mathscr{D}(X)$ is a lot more involved, and you cannot do this (natural) approach there due to the failure of Zariski descent for $h\mathscr{D}(X)$.
The last example is algebraic K-theory. You cannot define the higher algebraic K-theory of a triangulated category (only the $K_0$), there are different stable ∞-categories with the same underlying triangulated category but different algebraic K-theory spaces. As someone who works a lot on algebraic K-theory, this for me is enough of a drawback to make me not wanting to use triangulated categories.
