We study transport across either a potential or a magnetic barrier which is placed on the top surface of a three-dimensional thin topological insulator (TI). For such thin TIs, the top and bottom surfaces interact via a coupling lambda which influences the transport properties of junctions constructed out of them. We find that for junctions hosting a potential barrier, the differential conductance oscillates with the barrier strength. The period of these oscillations doubles as the coupling lambda changes from small values to a value close to the energy of the incident electrons. In contrast, for junctions with a magnetic barrier, the conductance approaches a nonzero constant as the barrier strength is increased. This feature is in contrast to the case of transport across a single TI surface where the conductance approaches zero as the strength of a magnetic barrier is increased. We also study the spin currents for these two kinds of barriers; in both cases, the spin current is found to have opposite signs on the top and bottom surfaces. Thus this system can be used to split applied charge currents to spin currents with opposite spin orientations which can be collected by applying opposite spin-polarized leads to the two surfaces. We show that several of these features of transport across finite width barriers can be understood analytically by studying the delta-function barrier limit. We discuss experiments which may test our theory.