* The margins of error in its measurements have been significantly reduced. For a variety of reasons, moreover, these may be the most precise data points available for these physical constants for a long time.
* A low value for the Hubble constant remains. This had been in tension with prior data but systemic errors in the prior data have been discovered that reduce that tension.
* A model with three rather than four light neutrinos is now favored at almost a five standard deviation level. We know from particle physics experiments that there are at least three light neutrinos. This rules out a fourth generation light neutrino or extra light sterile neutrino to explain a "reactor" anomaly which isn't looking like such an anomaly anymore anyway. This also tends to disfavor axions.
* The sum of the three neutrino masses is bounded to less than 160 meV at a 95% confidence level down from 230 meV after the first release of data. The best fit value is lower. The minimum value is about 100 meV in an inverted hierarchy of neutrino masses and about 59 meV in a "normal hierarchy" of neutrino masses. Thus, Planck still can't distinguish between the two scenarios, although science is getting very close to being able to do so. A degenerate neutrino hierarchy, where all three neutrino types have roughly equal masses has been ruled out.
Even though our knowledge of the absolute neutrino mass values isn't all that precise on a percentage basis (it is only slightly less precise than the accuracy with which we know the up and down quark masses), on an absolute magnitude of error basis (where the error range is +/- 50 meV), the neutrino masses are actually, by far, the most precisely known of the Standard Model fundamental fermion masses.
* There is still no data on primordial B-modes which the BICEP experiment proclaimed in tension with the old Planck data in a result that has been cast in doubt due to systemic error issues. This should come in the next few weeks, however. Thus, the question of the nature of cosmological inflation, if any, remains unresolved for the moment. Still, presentations like this one (see page 8) suggest that Planck will largely contradict and rule out the BICEP result, and greatly constrain inflation models to a narrow subset with little or no tensor component. The best fit is to a tensor mode r=0, and it is constrained to be less than r=0.09 v. BICEP's estimate of r=0.20.
Bottom line: The polarization data favors pretty much the most boring, most hostile to beyond the Standard Model physics result possible. The very simple six parameter lamdaCDM model remains a very good fit to the data.
The results also greatly constrain thermal dark matter models. As I explained in a comment on the dark matter link (links added in this post to original comment which lacks links):
Thus, Planck seems to exclude entirely thermal relic warm dark matter with a WIMP-like annihilation cross-section.
Given the multiple problems with thermal relic cold dark matter with a WIMP-like annihilation cross-section pointed out by Warm Dark Matter proponents, the combined exclusion would seem to rule out all thermal relic dark matter with a WIMP-like annihilation cross section.
The lambda CDM model (which uses a definition of Cold Dark Matter that includes both thermal relic WDM and thermal relic CDM), requires a thermal relic.
So, if there is thermal relic dark matter it must not have weak force annihilation cross-sections and must instead be truly sterile with respect to the weak, strong and EM forces, although it might have self-interactions that do not lead to DM annihilation or at a different cross-section.
CDM in the absence of self-interaction doesn't seem to work at all to fit the galaxy scale data, so at this point you either have thermal relic WDM as keV scale sterile fermions that interact only via gravity and fermi contact forces, or there is no thermal relic DM at all. Such particles wouldn't be produced at the LHC or would be to light to detect in current direct dark matter detection experiments.
Other studies also place very strict bounds on purely bosonic DM (also here).
Axions escape this issue because it is not a thermal relic form of DM, but have other issues.
Of course, none of this rules out a fifth force or force modification approach.
Lyman alpha forest data from Planck seem to rule out WDM with particles less than 3.3 keV in mass which are too heavy to solve other CDM problems.