Simply stated:
is the "cosmological constant problem" (the discrepancy of about 120 orders of magnitude between measured values of vacuum energy and the ones predicted by Quantum Field Theory) in any way related to the question of whether vacuum energy/cosmological constant is the Dark Energy responsible for the acceleration of cosmic expansion?
The answer is yes: they are related. The quantum field theory estimate is wrong by some huge factor (e.g. 120 orders of magnitude) but this could be avoided if it were possible to simply assert that the cosmological constant $\Lambda$ is zero. But the astrophysical observations suggest $\Lambda$ is not zero.
In a bit more detail, the situation is as follows. The observations of distant supernovas, combined with further evidence from CMB and from the distribution of galaxies, together suggest that there is the component called dark energy in the equations for the large-scale evolution of the cosmos. The observations are consistent with the hypothesis that this dark energy component takes the form of a cosmological constant, by which we mean a term in the equation which is simply a constant multiplied by the metric tensor.
Now let's turn to quantum field theory. There are at least two ways to estimate a possible cosmological constant term in any quantum field theory. In one way, one simply introduces a term in the Lagrangian which takes the form of a constant multiplied by the metric tensor. In the other way, one sums the zero-point energy of all the modes of the field up to some frequency of the order of the Planck frequency. Either calculation gives an answer too large by a huge factor (e.g. 120 orders of magnitude), unless one specifies an artificially low value for the constant in the Lagrangian method---but we don't know why it should be low. If one could assert that the constant is in fact zero, then the problem would be much reduced, since our assessments of mathematical beauty suggest that leaving a term completely out of an equation tends to preserve (or even enhance) the sense of harmony or coherence in the ideas captured in the equation. But it is hard to say why there would be a non-zero value with a size of order $10^{-120}$. If the constant is there at all, then one would expect it to have a value of order $1$ in the absence of some reason or mechanism to suppress it.
