Residential battery energy storage systems (BESS) and dynamic electricity tariffs are important building blocks for grid integration of renewable electricity. While various BESS operation strategies have been studied, a systematic comparison under dynamic tariffs (real-time pricing) is missing. Using rule-based and linear optimization approaches grounded on 5 years of data from 448 households, we estimate upper and lower bounds of net gains from using fixed and dynamic pricing. For a 10 kWh BESS, dynamic tariffs at a similar price level as fixed tariffs yield 12.7 % higher net gains, even under a price-agnostic self-consumption maximization strategy. We find that a perfect day-ahead forecast increases net profits by 6 % compared to naïve forecasts (e.g., previous day / week). Compared to a simple rule-based approach, operating BESS with linear optimization over a full-year of perfect information generates additional net profits of 14 %. With day-ahead forecasting, BESS above 15 kWh are best operated by maximizing self-consumption. Our findings offer practical guidance for homeowners considering investments and demonstrate benefits of switching tariffs. They also support policymakers in designing subsidies and regulations for private BESS and dynamic tariff adoption. For utility companies, the results help advance tariff schemes to redistribute market risks and realize grid flexibilities.