Why do we need effective action $\Gamma$ given the connected generating functional $W$? I have just learnt the path integral formalism in QFT, up to the point where we computed the generating functionals $\mathcal{Z}[J] := Z[J]/Z[0]$, $W[J]$, and $\Gamma[\varphi]$. Here $J(x)$ is the classical current and $\varphi(x)$ is defined to be the functional derivative $\delta W[J]/\delta J$.
All the resources I perused (standard textbooks, lecture notes) mostly detail very carefully how to compute or derive them. What I don't understand is why do we still need effective action if we already have $W[J]$. The sum of connected diagrams already gives everything you need in perturbation theory (in fact you don't really need $Z[J]$ anymore if you have $W[J]$). Furthermore, we often assume (at least in simple examples, since I haven't reached gauge theory) that the Legendre transformation of $W[J]$ is involutive, so we do not lose information working with either $\Gamma$ or $W$.
Since we have to do Feynman diagrammatics anyway, I doubt the reason we prefer one of them or the other is because one of them has easier Feynman diagrams integrations. So either (1) somehow the effective action trades off something for some advantages (that I cannot appreciate) over the connected generating functional $W[J]$, or (2) there's something about semi-classical vs quantum correlators that I don't understand: i.e. maybe $W[J]$ makes no sense for classical field theory but $\Gamma$ somehow does.
I would appreciate an explanation and/or explicit example that $\Gamma$ is absolutely preferred over computing $W$.
 A: It seems like you're looking for an answer along the lines of "you always want to compute quantity $x$ because reasons $y$ and $z$," but there's no such answer. All of these quantities are useful in different contexts, so it really depends on what you're doing. You say that $W[J]$ already gives us everything because the coefficients in its $J$ function power series are the connected correlation functions, but this seems like the bias of someone who is only interested in computing scattering amplitudes via the LSZ reduction.
Effective field theory itself works with the quantity $\Gamma[\phi]$ since $\phi$ (which is really a misleading shorthand for the tadpole $\langle\phi\rangle$) satisfies equations $\frac{\delta\Gamma}{\delta\phi}=0$ for zero current. Hence $\Gamma$ provides the equations of motion for the tadpoles, and if you were looking for quantum corrections to the classical theory, $\Gamma$ is really the object you're interested in. Hence effective field theory.
I would also point out that $\Gamma$ is the correct object to consider when thinking about spontaneous symmetry breaking. It's standard to say that the breaking is given by having a minima in the classical potential, but this is actually only the tree level result. Strictly speaking, you need to look for minima in the potential of the effective action, since that's what's determining $\langle\phi\rangle$. For example, Coleman and Weinberg worked out at some point the 1-look correction you get to spontaneous symmetry breaking (in QED if I remember correctly).
There is also an intimate (diagramatic) relationship between $W$ and $\Gamma$ which is quite nice, but for that I'll refer you to the QFT book by Banks: "Modern Quantum Field Theory: A Concise Introduction." He performs essentially no calculations in the text, leaving all of them as exercise problems, but it's still a good book for picking up some of the modern ways of thinking about concepts in QFT, leaving the details for other sources. For example, the QFT book by Nair will also contain more modern thoughts about QFTs with the level of detail you might expect from a textbook, but as a result it is a much more significant undertaking to read.
A: The effective action $\Gamma[\phi_{\rm cl}]$ is the generator of 1-particle irreducible (1PI) diagrams, which means than there are fewer Feynman diagrams to calculate than in the generator $W_c[J]$ of connected diagrams, something a practitioner of QFT will greatly appreciate. In other words, it is a more efficient organizing principle for doing QFT calculations.
