Each period is 2 weeks. First week is competition, and second week is reward week. The winning side geets entrance to catacombs, necropolices and trading rights with mamons. Loosing side can't get to the dungens and get a debuff in towns. Oh, and if you turned in sealstones you can only get them if your's side won.
Both sides can enter a necroplis or catacomb under certain conditions, even during event week. It has happened once during my two years playing L2. If, for instance, Dawn side wins Seal of Avarice, but does not win Seal of Gnosis, then any player may enter a catacomb during reward week. Only necropoli are restricted. Or if They won Gnosis, but did not break Avarice, then all players would have access to necropoli, but only the winning side could enter catacombs.