Difference between 1PI effective action and Wilsonian effective action?

The Wilsonian effective action is an action with a given scale, where all short wavelength fluctuations (up to the scale) are integrated out. Thus the theory describes the effective dynamics of the long wavelength physics, but it is still a quantum theory and you still have an path integral to perform. So separating the fields into long and short wavelength parts $phi = phi_L + phi_S$, the partition function will take the form (N.B. I'm using euclidean path integral)

$$ Z = intmathcal Dphi e^{-S[phi]} =intmathcal Dphi_{L}left(int Dphi_{S}e^{-S[phi_L+phi_S]}right)=intmathcal Dphi_{L}e^{-S_{eff}[phi_L]}$$ where $S_{eff}[phi_L]$ is the Wilsonian effective action.

The 1PI effective action doesn't have a length scale cut-off, and is effectively looking like a classical action (but all quantum contribution are taken into account). Putting in a current term $Jcdot phi$ we can define $Z[J] = e^{-W[J]}$ where $W[J]$ is the generating functional for connected correlation functions (analogous to the free energy in statistical physics). Define the "classical" field as $$Phi[J] = langle 0|hat{phi}|0rangle_J/langle 0| 0 rangle_J = frac 1{Z[J]}frac{delta}{delta J}Z[J] = frac{delta}{delta J}left(-W[J]right).$$

The 1PI effective action is given by a Legendre transformation $Gamma[Phi] = W[J] + JcdotPhi$ and thus the partition function takes the form

$$Z = intmathcal D e^{-S[phi] + Jcdot phi} = e^{-Gamma[Phi] + Jcdot Phi}.$$ As you can see, there is no path integral left to do.