Direct comparison of the magnetic and transport critical current density (Jc) for the same pieces of copper-sheathed MgB2 wires shows a large discrepancy in magnitude and field dependence of the two. The value of magnetic Jc can differ from the value of transport Jc by a factor of 10 or more. This discrepancy does not occur merely because of the difference in the voltage at which the magnetic and transport Jc are measured, but mainly because of the specific microstructure of MgB2. Such microstructure results in superconducting screening on at least two different length-scales, despite the absence of weak links in MgB2, leading to erroneous magnetic Jc if a simple critical state model is applied to such a system. Consequently, the magnetic Jc cannot be used for analysis of physical processes where the accurate field dependence of Jc needs to be known, such as vortex pinning. The magnetic Jc can still be used for qualitative comparison of MgB2 samples if they are all of the same size and have the same microstructure, or if their size is large enough so that the magnetic Jc does not depend on the sample size.