Conventional density functional theory (DFT) is strictly limited to describing the electronic density of ground electronic states. This is because the Hohenberg–Kohn theorems, on which it is based, are restricted to non-degenerate ground states (with no magnetic field). Also, because DFT aims at solving the time-independent Schrödinger equation (i.e. it explicitly finds stationary solutions), time-dependent phenomena are not accessible.
From the ground state density, any property of the system including excited state properties could in theory be derived given an appropriate functional $X[\rho]$ that maps the ground state density $\rho$ onto the property $X$. However, in practice, no known functional exists that maps the ground state density to excitation energies or other excited-state properties. Thus, excited states and their properties are in practice unreachable by DFT: I suppose this is what is meant by Gross when stating them as “notoriously difficult to calculate”. I myself would have said impossible.
Finally, to give a short list of properties that can be described with TDDFT but not with DFT — are thus all excited state properties. In particular: bandplots/bandgaps, excitation energies, transition moments, photo absorption spectra, frequency-dependent response properties (optical & dielectric properties), etc.
Specifically, one has to be careful about band plots and band gaps, which are routinely reported from time-independent DFT calculations: they are a representation of the Kohn–Sham energies, i.e., the energies of the fictive non-interacting Kohn–Sham system, which has no physical interpretation at all.
 Note here that we are talking about a functional in the generic, mathematical meaning of the word. It is not the exchange–correlation functional.