We study turbulence on closed and open flux surfaces in a comparative manner using the three-dimensional electromagnetic gyrofluid turbulence code GEM. A magnetic field on a tokamak is doubly periodic and sheared. This leads to the so-called field line connection, which ensures a finite parallel response for every degree of freedom. In contrast, in the scrape-off layer (SOL), the field lines end on plates, breaking this constraint and allowing the existence of convective cell modes. Since the parallel electron response provides a path to dissipation, whether or not it is allowed to vanish is important. For the SOL case, a standard Debye sheath model is used to provide the parallel boundary conditions. A zero loss model (no fluxes into the plates) is also used to assess the importance of the Debye currents. Turbulence on closed and open flux surfaces at the same parameters is found to be very diifferent, a property which basic transport models should take into account.