To expand on user1420303's answer a bit:
When I want to compare the energy between two different spin states of the same molecule, i.e. singlet-triplet-gap, do I better use unrestricted or restricted open formalism to compare the energies between both?
The unrestricted formalism will almost always give lower absolute electronic energies (i.e., closer to the "real" value) than restricted-open will, due to the greater variational flexibility introduced by the separate spin-up and spin-down orbital sets. So, from a purely energetic perspective, yes, unrestricted is likely preferred.
If you're interested in other properties, though, the spin contamination introduced by the unrestricted formalism may be problematic. Of course, in systems with sufficient static correlation (viz., a degenerate or near-degenerate ground state) to result in spin contamination (radicals, bond-breaking states, transition metals, etc.), single-reference methods like Hartree-Fock and DFT may not give reliable results anyways.
Further on, do I need to calculate the singlet state also with restricted open/unrestricted method?
For most 'non-exotic' organic systems, there is no need to calculate the singlet state using an unrestricted method since, as you note, the results will often be indistinguishable. For systems with static correlation, though, if you still want to try to use a single-reference method, you will need to explore the unrestricted orbital space, as there may be a lower-energy solution where the alpha and beta orbital compositions appreciably differ. The primary method I'm aware of for such exploration is the "broken-symmetry" method. Two papers I know of that discuss the method, albeit not in extensive detail, are the following reviews by Frank Neese:
F. Neese. "Prediction of molecular properties and molecular spectroscopy with density functional theory: From fundamental theory to exchange-coupling." Coord Chem Rev 253: 526, 2009. doi:10.1016/j.ccr.2008.05.014
F. Neese. "A critical evaluation of DFT, including time-dependent DFT,
applied to bioinorganic chemistry." J Biol Inorg Chem 11: 702, 2006. doi:10.1007/s00775-006-0138-1
The method is integrated into the current version of ORCA (see Section 5.9.10 of the v3.0.3 ORCA manual), and tips for running such calculations are available at the ORCA Input Library. Other software packages likely include this capability; search their manuals for more information.