Currently, the statistics on COVID-19 for many regions are accumulated for the time span of over than two years, which facilitates the use of data-driven algorithms, such as neural networks, for prediction of the disease's further development. This article provides a comparative analysis of various forecasting models of COVID-19 dynamics. The forecasting is performed for the period from 07/20/2020 to 05/05/2022 using statistical data from the regions of the Russian Federation and the USA. The forecast target is defined as the sum of confirmed cases over the forecast horizon. Models based on the Exponential Smoothing (ES) method and deep learning methods based on Long Short-Term Memory (LSTM) units were considered. The training data set included the data from all regions available in the full data set. The MAPE metric was used for model comparison, the evaluation of the effectiveness of LSTM in the learning process was carried out using cross-validation on the mean squared error (MSE) metric. The comparisons were made with the models from various literature sources, as well as with the baseline model "tomorrow as today" (for which the sum of cases over the forecast horizon is supposed to be equal to the current case number multiplied by the forecast horizon length). It was shown that on small horizons (up to 28 days) the "tomorrow as today" model and ES algorithms show better accuracy than LSTM. In turn, on longer horizons (28 days or more), the preference should be given to the more complex LSTM-based model.