Climate change impact studies are essential for understanding the effects on water resources under changing climate conditions. This paper assesses the effectiveness of Long Short-Term Memory (LSTM) neural networks versus traditional hydrological models for these studies. Traditional hydrological models, which rely on historical climate data and simplified process parameterization, are scrutinized for their capability to accurately predict future hydrological streamflow in scenarios of significant warming. In contrast, LSTM models, known for their ability to learn from extensive sequences of data and capture temporal dependencies, present a viable alternative. This study utilizes a domain of 148 catchments to compare four traditional hydrological models, each calibrated on individual catchments, against two LSTM models. The first LSTM model is trained regionally across the study domain of 148 catchments, while the second incorporates an additional 1,000 catchments at the continental scale, many of which are in climate zones indicative of the future climate within the study domain. The climate sensitivity of all six hydrological models is evaluated using four straightforward climate scenarios (+3 °C, +6 °C, -20 %, and +20 % mean annual precipitation), as well as using an ensemble of 22 CMIP6 GCMs under the SSP5-8.5 scenario. Results indicate that LSTM-based models exhibit a different climate sensitivity compared to traditional hydrological models. Furthermore, analyses of precipitation elasticity to streamflow and multiple streamflow simulations on analogue catchments suggest that the continental LSTM model is most suited for climate change impact studies, a conclusion that is also supported by theoretical arguments.