Excited-states properties with traditional density functional theory
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.
pH13 - Yet another Philipp about 3 years
Recently I was shown the book "Density Functional Theory II, Relativistic and Time Dependent Extensions" by Nalewajski in which a chapter from E. K. U. Gross et al. is printed.  This chapter contains an interesting statement in it's introduction:
Excited-state properties, however, are notoriously difficult to calculate within the traditional density functional framework and time-dependent phenomena are not accessible at all.
What are those excited-state properties that are "notoriously difficult to calculate" and what makes it hard to calculate them without the TDDFT framework?
 Gross, E. K. U.; Dobson, J. F.; Petersilka, M. Density Functional Theory of Time-Dependent Phenomena. In Density Functional Theory II; Nalewajski, R. F., Ed.; Topics in Current Chemistry 181; Springer Berlin Heidelberg, 1996; pp 81–172.
Ben Norris about 7 yearsWelcome to Chemistry.SE! This answer is basically a 'link-only' answer. If the content the link points to changes or is removed, this answer becomes worthless. It also points to an article that might be behind a paywall for some users. Feel free to improve your answer by summarizing some of the key points of the article you linked.