Artificial intelligence (AI)-based approaches are crucial in computer-aided diagnosis (CAD) for various medical applications. Their ability to quickly and accurately learn from complex data is remarkable. Deep learning (DL) models have shown promising results in accurately classifying Alzheimer's disease (AD) and its related cognitive states, Early Mild Cognitive Impairment (EMCI) and Late Mild Cognitive Impairment (LMCI), along with the healthy conditions known as Cognitively Normal (CN). This offers valuable insights into disease progression and diagnosis. However, certain traditional machine learning (ML) classifiers perform equally well or even better than DL models, requiring less training data. This is particularly valuable in CAD in situations with limited labeled datasets. In this paper, we propose an ensemble classifier based on ML models for magnetic resonance imaging (MRI) data, which achieved an impressive accuracy of 96.52%. This represents a 3-5% improvement over the best individual classifier. We evaluated popular ML classifiers for AD classification under both data-scarce and data-rich conditions using the Alzheimer's Disease Neuroimaging Initiative and Open Access Series of Imaging Studies datasets. By comparing the results to state-of-the-art CNN-centric DL algorithms, we gain insights into the strengths and weaknesses of each approach. This work will help users to select the most suitable algorithm for AD classification based on data availability.