Hi and welcome to Cross Validated!
The difference between the MLE and QML is rather subtle, but it has to do with the assumption of normality and its appropriateness. In the GARCH setting, the assumption of normality of returns is particularly useful because it significantly simplifies calculations in the likelihood function. However, it is also a "well-established" fact that financial returns are rarely normally distributed, even over longer time horizons when one would expect the Central Limit Theorem effects to kick in.
To quote McNeil, Frey & Embrechts themselves, from their book Quantitative Risk Management － Concepts, Techniques and Tools, Revised edition, on section 4.2.4 Fitting GARCH Models to Data, p. 146:
In describing the behaviour of parameter estimates in the following paragraphs,
we distinguish two situations. In the first situation we assume that the model that
has been fitted has been correctly specified, so that the data are truly generated by a
time-series model with both the assumed dynamic form and innovation distribution.
We describe the asymptotic behaviour of the maximum likelihood estimates (MLEs)
under this idealization.
In the second situation we assume that the correct dynamic form is fitted but that
the innovations are erroneously assumed to be Gaussian. Under this misspecification,
the model fitting procedure is known as quasi-maximum likelihood (QML) and the
estimates obtained are QMLEs. Essentially, the Gaussian likelihood is treated as an
objective function to be maximized rather than a proper likelihood; our intuition
suggests that this may still give reasonable parameter estimates, and this turns out
to be the case under appropriate assumptions about the true innovation distribution.
In other words, you can use the typical ML when you know that your $z_q$ do indeed follow a normal distribution. Otherwise, when you can see, for example via appropriate tests, that your $z_q$ are not normally distributed, you should use QML (as long as the innovations are iid with zero mean and unit variance).