Depression is a common mental disorder that negatively affects physical health and personal, social and occupational functioning. Currently, accurate and objective diagnosis of depression remains challenging, and electroencephalography (EEG) provides promising clinical practice or home use due to considerable performance and low cost. This work investigates the capabilities of deep neural networks with EEG-based neural patterns from both resting states and cognitive tasks for depression detection. We collect EEG signals from 33 depressed patients and 40 healthy controls using wearable dry electrodes and build Attentive Simple Graph Convolutional network and Transformer neural network for objective depression detection. Four experiment stages, including two resting states and two cognitive tasks, are designed to characterize the alteration of relevant neural patterns in the depressed patients, in terms of decreased energy and impaired performance in sustained attention and response inhibition. The Transformer model achieves an AUC of 0.94 on the Continuous Performance Test-Identical Pairs version (sensitivity: 0.87, specificity: 0.91) and the Stroop Color Word Test (sensitivity: 0.93, specificity: 0.88), and an AUC of 0.89 on the two resting states (sensitivity: 0.85 and 0.87, specificity: 0.88 and 0.90, respectively), indicating the potential of EEG-based neural patterns in identifying depression. These findings provide new insights into the research of depression mechanisms and EEG-based depression biomarkers.