Influence of galactic cosmic rays (GCRs) on cloud formation is suggested to be an important part of the mechanism of solar activity influence on weather and climate. A high positive correlation between low cloud amount and GCR fluxes was observed in the 1980s-1990s; however, in the early 2000s, it was violated. In this work, we consider a nature of long-term correlation links between cloud cover at middle latitudes and GCRs, as well as possible reasons for this correlation reversal. It was shown that the GCR-cloud links observed on the decadal time scale are indirect and caused by GCR effects on cyclonic activity which depend on epochs of the large-scale atmospheric circulation. The reversal of GCR-cloud correlation in the 2000s seems to be due to a sharp weakening of the Arctic and Antarctic stratospheric polar vortices, which results in the change of the troposphere-stratosphere coupling and, then, of GCR contribution to the development of extratropical cyclogenesis.