Your question might be compacted to someting like: Do I need derived categories to study cohomology of sheaves? Of course, the answer depends on your particular interests. Let me anyway give you some starting points to help you to make up your mind.
(1) Are the derived categorical derived functors universal in the classical sense?
Let $f : \mathcal{A} \to \mathcal{B}$ a left exact functor between abelian categories and denote also by $f$ its extension to the corresponding homotopy categories. $\mathbf{K}(\mathcal{A})$ denotes the category of complexes with maps up to homotopy. The derived functor $\mathbf{R}f : \mathbf{D}(\mathcal{A}) \to \mathbf{D}(\mathcal{B})$ satisfies a universal property that implies that the collection $\{\mathbf{R}^if\}_{i \in \mathbb{N}}$ is a universal $\delta$-functor such that $\mathbf{R}^0f = f$. Where $\mathbf{R}^if: = \mathrm{H}^i\mathbf{R}f$. See [L, $\S2.1$].
(2) Can we show the "Grothendieck's Tohoku" easily using derived category?
This is related to the existence of acyclic objects for a certain functor. For details, I suggest you to look at [L, $\S$ 2.2].
(3) How can we use derived categorical derived functors in order to show propositions around spectral sequences?
As long as you refer to the so called Grothendieck spectral sequence on the composition of functors, it is replaced by the following theorem. Let $f : \mathcal{A} \to \mathcal{B}$ and $g : \mathcal{B} \to \mathcal{C}$ left exact functors between abelian categories. If $f$ takes $f$-acyclic objects into $g$-acyclic objects we have a natural isomorphism
$$
\mathbf{R}gf \cong \mathbf{R}g\mathbf{R}f
$$
(there is an analogous theorem for left derived functors).
Thus, every time $\mathbf{R}f$ reduces to $f$ you obtain similar formulas than the ones you obtain by the collapse of the spectral sequence. The advantage is that the argument is simpler and you don't have limitations on finiteness or boundedness of the complex involved.
Also, arguments involving three or more functors are seamless, something that would require multi-graded spectral sequences.
On the other hand, derived categories will never help you in computing delicate properties for certain spectral sequences like the Adams spectral sequence or similar ones.
(4) Grothendieck group of a derived category
In favorable cases $\mathrm{K}_0(\mathbf{D}(\mathcal{A}))$ and $\mathrm{K}_0(\mathcal{A})$ agree. A small advantage would be that the oposite of a class of an object $X$ is not a virtual object but $-[X] = [X[1]]$. A discussion of $\mathrm{K}_0$ in the geometric context is in SGA6, exposé IV, to begin with.
(5) Easier proof of classical homological propositions
I won't assert they are easier but they are in my opinion clearer and broader. For base-change and Künneth I suggest you to look at [L, Theorem (3.10.3)]. The Künneth formula is more general than any other I've seen in the literature. Its expression via spectral sequences, if possible, would look extremely complicated.
Besides, if you really want to understand Grothendieck-Serre duality beyond Cohen-Macaulay maps and schemes, then derived categories are indispensable.
Final remarks
For me, [L] is a very good introduction to the use of derived categories in Algebraic Geometry. Unfortunately it is not self-contained, so you need at least the first chapter in [KS1] and looking at Spaltenstein paper on unbounded resolutions. Alternatively you have all the needed prerequisites in [KS2].
Bibliography
[KS1] Kashiwara, Masaki; Schapira, Pierre: Sheaves on manifolds. Grundlehren der Mathematischen Wissenschaften, 292. Springer-Verlag, Berlin, 1994.
[KS2] Kashiwara, Masaki; Schapira, Pierre: Categories and sheaves. Grundlehren der Mathematischen Wissenschaften, 332. Springer-Verlag, Berlin, 2006.
[L] Lipman, Joseph: Notes on derived functors and Grothendieck duality. Foundations of Grothendieck duality for diagrams of schemes, 1–259, Lecture Notes in Math., 1960, Springer, Berlin, 2009.
[SGA6] Berthelot, Pierre; Alexandre Grothendieck; Luc Illusie, eds. Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics 225). Berlin; New York: Springer-Verlag, 1971.
Best Answer
Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories with enough injective objects. Let me use the notation $D^+(\mathcal{A})$ and $D^+(\mathcal{B})$ to denote the stable $\infty$-categories whose homotopy categories are the (cohomologically bounded below) derived categories of $\mathcal{A}$ and $\mathcal{B}$, respectively (you can also consider unbounded derived categories, but the situation is a bit more subtle).
Let $\mathcal{C} \subseteq \mathrm{Fun}( D^{+}( \mathcal{A} ), D^{+}( \mathcal{B}) )$ be the full subcategory spanned by those functors which are exact, left t-exact, and carry injective objects of $\mathcal{A}$ into the heart of $D^{+}( \mathcal{B} )$. Then the construction $$F \in \mathcal{C} \mapsto h^0 F|_{ \mathcal{A} }$$ determines an equivalence from $\mathcal{C}$ to the category of left exact functors from $\mathcal{A}$ to $\mathcal{B}$. The inverse of this equivalence is "taking the right derived functor".
Consequently, one can answer your question as follows: given a functor of triangulated categories $G: hD^{+}(\mathcal{A}) \rightarrow hD^{+}(\mathcal{B})$, it arises as a right derived functor (of a left exact functor of abelian categories) if and only if
a) The functor $G$ lifts to an exact functor of stable $\infty$-categories $D^{+}(\mathcal{A}) \rightarrow D^{+}(\mathcal{B})$ (anything that you build by composing derived functors will have this property).
b) The functor $G$ is left t-exact and carries injective objects of $\mathcal{A}$ into the heart of $hD^{+}(\mathcal{B})$.