It is hard to know where to begin! A general principle is that as long as you are only concerned with the derived category of a single variety, it is generally sufficient to consider it as a triangulated category; while as soon as you are interested in families of derived categories, it becomes impossible to stay in the triangulated setting, so that it becomes necessary to pass to some homotopical refinement. The basic reason for this is the extremely poor behaviour of the homotopy theory of triangulated categories.

The most striking example is the work of Orlov on Fourier-Mukai functors. A triangulated functor between derived categories of schemes is called of Fourier-Mukai type if it of geometric origin, i.e. if it is given by twisting by a complex on the product of the schemes. Orlov showed in the 90's that for smooth projective varieties over a field, any fully faithful functor between derived categories of coherent sheaves is of Fourier-Mukai type. It was expected, or hoped, that every functor is Fourier-Mukai, but this turned out to be impossible to prove in the setting of triangulated categories, literally: a counter-example was finally found last year by Van den Bergh and Rizzardo (arXiv:1410.4039), while the dg-categorical analogue was proved ten years ago by Bertrand To\"en (arXiv:0408337). A brief glance at the latter paper will be enough to convince oneself that the proof relies heavily on studying the homotopy theory of dg-categories.

A second important application is algebraic K-theory. At the triangulated level, the only part of the algebraic K-theory of $X$ that can be defined from only the triangulated structure is the Grothendieck group. However, it has been believed that the algebraic K-theory of a scheme can be recovered from its derived category for a long time (in the sense that, if two schemes have triangulated-equivalent derived categories, then their higher K-theory groups are isomorphic). Neeman spent many years working on this problem, and I believe that in the end, his work implies that this is true under some assumptions like regularity. However, the proof takes up eight papers of around one hundred pages each. On the other hand, after work of Schlichting, To\"en and others, it is possible to define the algebraic K-theory of a dg-category in such a way that it recovers Thomason-Trobaugh K-theory for the dg-derived category of a scheme. In fact, one can deduce from this the analogous result for triangulated categories: using the theory of Fourier-Mukai functors mentioned above, it is possible to prove that the derived categories of two schemes (quasi-compact, separated over a field, if I recall correctly) are triangulated-equivalent iff their dg-derived categories are quasi-equivalent.

Related to this is the question of descent. As bananastack mentioned, the derived category satisfies Zariski descent only at the dg-level. In fact, at the dg-level it even satisfies Nisnevich descent. I believe the latter fact is due independently to Bhatt, Drinfeld and Lurie. From this it is possible to deduce a nice proof of the theorem of Thomason-Trobaugh on Nisnevich descent for K-theory. The missing ingredient is the localization fibre sequence, which can be deduced from compact generation properties of the derived category, which can in turn be proved in an elegant way at the dg-level, as explained in the beautiful paper arXiv:1002.2599 of To\"en.

Finally, let me note that, even though the various dg-enhancements constructed in the literature seem rather complicated, the dg-categorical derived category is in fact a more natural object than the triangulated one, in the following sense. The reason one considers the derived category of $X$ in the first place is in order to talk about "homotopy theory of chain complexes of coherent sheaves on $X$, up to quasi-isomorphism". The naive way to make this precise is to take the Gabriel-Zisman localization of the category of chain complexes at the class of quasi-isomorphisms, and then manually identify the distinguished triangles. The smart way is to take the Dwyer-Kan simplicial localization instead, which is a homotopically correct version of Gabriel-Zisman localization which doesn't kill higher homotopies; more precisely, it is a simplicially enriched category whose simplicial Hom-sets have $\pi_0$ identified with the Hom-sets of the Gabriel-Zisman localization. This gives rise to a dg-enhancement where the distinguished triangles are already built in as the (co)fibre sequences.