I think the wikipedia page gives a reasonable overview of why gravitational waves give us a window on the universe that is either not observable or is complementary to the view we get from electromagnetic waves (or neutrinos).
But let's have a specific example. The potential of LIGO, as we have seen today, is to detect the gravitational wave signatures of merging black hole binaries.
The power radiated in gravitational waves strongly depends on the orbital separation of the black holes, it also depends on their masses. The frequency of the radiation will change as the black holes spiral together. The "noise" that this makes in the final few seconds as the black holes merge is the "chirp" that is detected by LIGO. Have a look (listen!) to what a chirp might "sound" like at this web page from University of Birmingham.
The exact characteristics of the chirp (duration, frequency etc.) can for instance yield the masses of the black holes and of their final configuration and may tell us about spin rates of the black holes. Yet these same mergers may produce zero electromagnetic wave signatures that could be seen by conventional telescopes.
Furthermore, because the amplitude and the frequency of the waves depend differently on the masses and distance to the GW source, these binary chirps acts as "standard candles" in a similar way to type Ia supernovae. In other words, measurements of the chirp independently gives the masses of the merging black holes and the distance to them.