Low frequency fluctuations (LFFs) of the BOLD signal are a major discovery in the study of the resting brain with functional magnetic resonance imaging (fMRI). Two fMRI-based measures, functional connectivity (FC), a measure of signal synchronicity, and the amplitude of LFFs (ALFF), a measure of signal periodicity, have been proved to be sensitive to changes induced by several neurological diseases, including degenerative dementia. In spite of the increasing use of these measures, whether and how they are related to each other remains to be elucidated. In this work we used voxel-wise FC and ALFF computed in different frequency bands (slow-5: 0.01-0.027 Hz; slow-4: 0.027-0.073 Hz; and full-band: 0.01-0.073 Hz), in order to assess their relationship in healthy elderly as well as the relevant changes induced by Alzheimer's Disease (AD) and Mild Cognitive Impairment (MCI). We found that in healthy elderly subjects FC and ALFF are positively correlated in anterior and posterior cingulate cortex (full-band, slow-4 and slow-5), temporal cortex (full-band and slow-5), and in a set of subcortical regions (full-band and slow-4). These correlation patterns between FC and ALFF were absent in either AD or MCI patients. Notably, the loss of correlation between FC and ALFF in the AD group was primarily due to changes in FC rather than in ALFF. Our results indicate that degenerative dementia is characterized by a loss of global connection rather than by a decrease of fluctuation amplitude.