Evidence of why the Standard Model is a successful theory of particle physics When discussing physics with laypersons, I'm often in the situation where I have to explain what the Standard Model is, and why it's a successful theory of particle physics.  
To help in such situations, I'm looking for explicit examples of where the Standard Model has been successful - i.e. the prediction of certain particles, incredible agreement between experimental/theoretical results and so on.
Conversely, what, if any, are the failures of the Standard Model (apart from the the obvious not-including-gravity one). Are there theoretical predictions which differ significantly from experimental results?
 A: Real physical theories are constructed on the ground of experiments. What I mean is that a model like the Standard Model of Particle Physics was not born as it is currently. There were lots of attempts, trial and error, etc.
The success of the current SM of particle physics is mainly because one can explain lots of experimental evidence within its framework.
I agree with PhotonicBoom that the prediction of the masses of the $W^{\pm}$ and $Z$ bosons is one of the major successes of the SM (or a hint of the validity of the Higgs mechanism if you prefer). 
However, the real potential of the SM comes with the so called precision measurements.


*

*The measured mass of the $Z$ boson agrees with the theoretical predictions (even when loop corrections are taken into account).

*The anomalous gyromagnetic moment of electron and muon: although there are still discrepancies of the later with the experimental data, the former gives the best agreement between theory and experiment.

I wouldn't say that SM fails in the giving masses to neutrinos, explaining the hierarchy problem, explaining dark matter or including gravity. Simply it is not meant to cover those topics:


*

*The neutrino oscillation occurs, i.e., neutrinos have masses. This feature can be included into the SM with no pain. I would even say that no one in the scientific community would say that the SM do not cover this topic. But we have no agreement (yet) on the mechanism of providing masses to neutrinos.

*Hierarchy problem: I don't believe it is a problem, it is a fact... nature has different scales! And the SM is not constructed for explaining this... it abides!

*Dark energy: Yes, it is there but there is no experimental evidence (in particle physics) that DE (or Dark Matter) are needed. Therefore, I'd argue that DM and DE have a gravitational origin (or course I'm probably wrong).

*Gravity: well, SM is constructed on the basis of a flat spacetime, so there is no intention of including gravity on its domain.
A: The lists will end up being huge, therefore I will only mention a few of each. This is my attempt of an answer: 
Successes of the Standard Model:


*

*Perhaps the biggest success of the Standard Model is the prediction of the Higgs Boson. The particle has been experimentally verified in 2012 (if my memory serves me well) after it has been theorised for over 50 years. 

*Other successes of the Standard model include the prediction of the W and Z bosons, the gluon, and the top and charm quark, before they have even been observed. 

*Another prediction also includes the anomalous magnetic dipole moment of the electron, which is given by $a = 0.001 159 652 180 73(28)$ which results in our most precise value of the fine structure constant: $α^{−1} = 137.035 999 070 (98)$, which is a precision of better than one part in a billion!

*Wikipedia has a table with the prediction of the masses of the W and Z boson compared with experimental data. It is evident that those are extremely accurate predictions:
$$
\begin{align*}
&\textrm{Quantity}&&\textrm{Measured (GeV)}&&\textrm{SM prediction (GeV)}\\
\hline
&\textrm{Mass of W boson}&&80.387\phantom0\pm0.019\phantom0&&80.390\phantom0\pm0.018\phantom0\\
&\textrm{Mass of Z boson}&&91.1876\pm0.0021&&91.1874\pm0.0021
\end{align*}
$$
Failures of the Standard Model:
The biggest one in my opinion is the complete absence of gravity in the SM. As you mentioned in your question though, you are interested in other failures, perhaps less known.
These include:


*

*The SM predicts neutrinos to be massless. We have observed neutrino oscillations which implies that neutrinos are massive (by massive I mean they have mass, there actual mass is tiny!).

*The Hierarchy Problem. In a nutshell, the SM cannot explain the large differences in the coupling constants of forces at low energy scales.

*The contribution of Dark Energy arising from the SM is many, many orders of magnitude higher than observed.

*CP violation in Cosmology
