Difference between discretization and quantization in physics
Quantisation does not imply discreteness. If a system has been quantised, we just mean we have taken the set of states, and replaced it by a vector space of states. In other words, one can add states in quantum mechanics, allowing a system to be in two states "at once". Observable quantities become certain operators acting on this vector space of states.
As you can see, this doesn't have anything, on the face of it, to do with discretisation. It turns out, however, that a lot of the operators we are interested in have discrete eigenvalues, and this implies that the corresponding physical values are discrete. Position, however, has a continuous spectrum, as do many other quantum observables.
There are plenty of sources explaining exactly how one goes from classical sets of states and numerical observables to quantum states (vector spaces - Hilbert spaces in particular) and quantum observables (operators); I won't cover that. All I shall say is that quantisation is a big mathematical process replacing a load of classical things with quantum things, and this sometimes leads to certain physical quantities being discretised.
Forget about language issues for a moment. The crucial feature of quantum mechanics is quantum-mechanical interference of probability amplitudes, which most people understand as a wave phenomenon (but may be described more elegantly and conceptually cleanly in terms of noncommutative matrix operations.) A mere consequence of that is that the spectra (sets of eigenvalues) of operators representing physical quantities (observables) are often, but not always, discrete. (Think back on wave equation spectra, etc.) The quantized energy spectra were the original puzzle that led physicists in the early 20th century to grip onto the discreteness, as classical mechanics for such quantities lacked it. That's why the name "quantum" was used, but other people working in the Schroedinger representation called it "wave mechanics".
By contrast, discretizing classical problems by putting them on a notional approximating lattice will, in general, not produce QM interference phenomena. Discretization is just a schematic numerical analysis approximation of continuous systems.
Now for your questions:
Quantization often relies on discreteness, so matrix methods of discrete mathematics are quite useful there, but its salient feature is QM interference of amplitudes, not just discrete structures. After all, even classical drum eigenvalue equations involve such discrete mathematical structures and quantized eigenvalues (and even interfering modes, but no probability waves). It is the bizarre probabilistic framework that makes QM.
It is fine to call the discrete part of a system's energy spectrum "discrete", but its continuous, scattering, part of course should be called "continuous", even though , it too is part of the quantum spectrum.
It is a terrible idea to mix these concepts. You'd invite stares, incomprehension, and discomfort.
In a nutshell, because it is not always discrete, and would not reflect the salient feature this answer starts with.
Aside : Angular momentum and spin are quantized observables in QM, and we do call them that. (Their quantization is not a convenience issue of numerical analysis: they behave in genuinely peculiar non-commutative ways.)
In quantum mechanics, bound states (i.e. energy eigenstates whose energy is less than the limiting value of the potential as $|{\bf x}| \to \infty$, whose wavefunctions therefore decay exponentially at large ${\bf x}$) typically have discrete energy levels, while scattering states (whose wavefunctions typically approach $e^{ikr}/r^\frac{d-1}{2}$ in $d$ spatial dimensions) have continuous energy spectra. So quantization doesn't necessarily imply discreteness.