Speech signals carry various bits of information relevant to the speaker such as age, gender, accent, language, health, and emotions. Emotions are conveyed through modulations of facial and vocal expressions. This paper conducts an empirical comparison of performances between the classical classifiers: Gaussian Mixture Model (GMM), Support Vector Machine (SVM), K-Nearest Neighbors (KNN), Artificial neural networks (ANN); and the deep learning classifiers, i.e., Long Short-Term Memory (LSTM), Convolutional Neural Network (CNN), and Gated Recurrent Unit (GRU) in addition to the ivector approach for a text-independent speaker verification task in neutral and emotional talking environments. The deep models undergo hyperparameter tuning using the Grid Search optimization algorithm. The models are trained and tested using a private Arabic Emirati Speech Database, Ryerson Audio–Visual Database of Emotional Speech and Song dataset (RAVDESS) database, and a public Crowd-Sourced Emotional Multimodal Actors (CREMA) database. Experimental results illustrate that deep architectures do not necessarily outperform classical classifiers. In fact, evaluation was carried out through Equal Error Rate (EER) along with Area Under the Curve (AUC) scores. The findings reveal that the GMM model yields the lowest EER values and the best AUC scores across all datasets, amongst classical classifiers. In addition, the ivector model surpasses all the fine-tuned deep models (CNN, LSTM, and GRU) based on both evaluation metrics in the neutral, as well as the emotional speech. In addition, the GMM outperforms the ivector using the Emirati and RAVDESS databases.