This study addresses the problem of Alzheimer's disease (AD) diagnosis with Electroencephalography (EEG). The use of EEG as a tool for AD diagnosis has been widely studied by comparing EEG signals of AD patients only to those of healthy subjects. By contrast, we perform automated EEG diagnosis in a differential diagnosis context using a new database, acquired in clinical conditions, which contains EEG data of 169 patients: subjective cognitive impairment (SCI) patients, mild cognitive impairment (MCI) patients, possible Alzheimer's disease (AD) patients, and patients with other pathologies. We show that two EEG features, namely epoch-based entropy (a measure of signal complexity) and bump modeling (a measure of synchrony) are sufficient for efficient discrimination between these groups. We studied the performance of our methodology for the automatic discrimination of possible AD patients from SCI patients and from patients with MCI or other pathologies. A classification accuracy of 91.6% (specificity = 100%, sensitivity = 87.8%) was obtained when discriminating SCI patients from possible AD patients and 81.8% to 88.8% accuracy was obtained for the 3-class classification of SCI, possible AD and other patient