Alzheimer's disease is a progressive neurodegenerative disease that has become one of the most central health issues in the world today. Early detection and proper diagnosis in the initial stages are essential for slowing disease progression and improving patient outcomes. This paper presents an integrated review and a practical evaluation of the most advanced machine-learning (ML) and deep-learning (DL) methods for diagnosing Alzheimer's disease (AD) in its initial stages, using neuroimaging data as input. As there is a possibility of better performance using classifier ensembles, which involve Support Vector Machine (SVM), Random Forest (RF), or Decision Tree (DT), or Logistic Regression (LR), or Naive Bayes (NB), or any combination of ensemble tactics, their diagnostic capabilities have been investigated. At the same time, Convolutional Neural Networks (CNNs), 3-D CNNs, Deep Neural Networks (DNNs), Feedforward Neural Networks (FFNNs), and hybrid CNN- LSTM models (CNN-LSTMs) have been evaluated in detail. The current state-of-the-art pretrained networks for transfer learning include VGG19, ResNet50, DenseNet, EfficientNetB0, MobileNet, Xception, and GapNet. Each model was trained on neuroimaging datasets and evaluated for accuracy, sensitivity, specificity, and area under the receiver operating characteristic curve (AUC-ROC). The results show that DL-based models improve their ability to identify subtle structural patterns in brain MRI scans compared to traditional ML models. In addition, ensemble ML models offer competitive performance and greater interpretability. This paper provides a comparative discussion of the merits, limitations, and practical challenges of the two methodology families (including overfitting, class imbalance, and generalizability) associated with both methods. The results are a step towards developing effective, automated, and innovative decision support tools for the early detection of Alzheimer's disease.