If we could build a telescope to view the cosmic neutrino background, what would we see? If we could build a neutrino telescope capable of viewing relic neutrinos that decoupled after the big bang, with a similar angular and spectral resolution that is possible now for the CMB (e.g. with Planck), what would we see
How would the C$\nu$B differ because of the finite neutrino mass and earlier decoupling? Would there be additional diagnostics and insights that are unavailable from the CMB? How big would the fluctuations in neutrino temperature be compared with the CMB? Would these fluctuations give us the neutrino mass or tell us something about the inflationary model?
NB: related questions are
Why are we blind for the era before the recombination?,
Seeing beyond the CMBR with neutrinos?
and
Is it possible to look into the beginning of the Universe?
but none these ask specifically, or have answers, about what could be seen or probed by the C$\nu$B if we could examine it in detail.
This question is somewhat hypothetical, but if you look at the existing (partial) answers and comments, you will see that there are technical developments that are making this more and more possible. 
 A: So over the last 48 hours I've been doing a small amount of research on my own question of why being able to detect or even image relic neutrinos would be important.
Perhaps the most succinct summary can be found in these slides  http://cosmo2014.uchicago.edu/depot/talk-long-andrew.pdf
These suggest that there are 3 types of answer:
The cosmologist's - that because neutrinos decouple only 1s after the big-bang, they push our understanding and test our model to more extreme limits than does the CMB ($\sim 10^{5}$ years) or even big-bang nucleosynthesis ($\sim 3$ minutes). The BB model suggests the background has a Fermi-Dirac distribution at 1.95K and there should be 56 neutrinos/cc (with equal numbers of anti-neutrinos) in the current Universe (modified by the effects below).
The particle physicist's - that the relic neutrino background should now be non-relativistic; the mean speed is
$$ v  \simeq 160 (1 +z)\ \left(\frac{{\rm eV}}{m_{\nu}}\right)\ \ {\rm km}\ {\rm s}^{-1},$$
where $z$ is the redshift and $m_{\nu}$ the neutrino mass,
and is subject to the influence of the gravitational fields of galaxy clusters (at z<2), the Milky Way and even the Sun. This modifies their phase space distribution away from a spatially uniform Fermi-Dirac distribution. Measurements of the background may tell us more about the neutrino mass(es), whether neutrinos decay, may discover sterile neutrinos, and may decide between Majorana and Dirac neutrinos. Simultaneously (or degenerately) it may tell us something about the history of the shape and size of the Milky Way dark matter halo. Ringwald & Wong (2004) also suggest that relic neutrino tomography might be possible, predicting over-densities of factors of 5 towards the Virgo cluster, for $m_{\nu}=$0.15 eV neutrinos, resolved with 2 degree resolution.
https://arxiv.org/abs/1404.0680
https://arxiv.org/abs/hep-ph/0408241
The experimentalist's - that if you can detect them, the Nobel prize is yours! It appears that despite some pessimistic comments, that the game is truly afoot - namely the PTOLEMY and KATRIN experiments aim to detect the C$\nu$B by neutrino capture by tritium nuclei (but not with spatial resolution!). It is a formidable challenge, with predicted event rates of $\sim 100$ year$^{-1}$ kg$^{-1}$ of Tritium target.
http://www.int.washington.edu/talks/WorkShops/int_10_44W/People/Formaggio_J/Formaggio.pdf
A: I will answer this since @rob, who provided the link that gives a summary of the proposed methods and technical difficulties, is not doing it (comments are not guaranteed to be invariant to time on this site).
It is true that measuring the Cosmic Microwave Background radiation has been extremely important in the development of  the model of the beginning of the universe called the Big Bang.. Relic radiation is radiation that has decoupled from the intense interactions that happen in the primordial soup of particles . In the case of photons, CMB, the decoupling happens 380.000 years after the BB.
This image gives an idea of the evolution of interactions  and decouplings in the primordial soup. To get such images one uses the whole panoply of knowledge from elementary particle physics interactions, the theoretical models that fit the data.

From this we see that the neutrino decouples at about 1 second, dependent of course on the calculations, as shown in this link. A great improvement of 380.000 years, and the relic distribution will carry information about the period before.
In this plot from the wiki article on BB 

We see that detecting relic neutrinos will give information for the development of the universe between 10^-32 seconds , which is the time the gravitational wave decoupled, to the time the neutrino decoupled. The BICEP2 experiment has managed to map the gravitational decoupling radiation in an ingenious way, using the polarization of CMB photons (the paper has been published). Thus, if we get the neutrino snapshot, we will see the evolution in time of the inhomogeneities that created the present density of superclusters of galaxies and clusters of galaxies. A consistent framework will increase the validity of  hypothesis entering the Big Bang model.
It will be important if experimenters succeed in detecting primordial neutrinos but as the firs link shows the technical difficulties are yet not surmounted, due to the very low energy of the relic neutrinos and their weak interaction with matter.  Neutrino physicists though are thinking about methods.
As Dmckee said in a comment, neutrino telescopes do exist but not for energies so much lower than sun neutrinos. 
