Calculating Stream Flow from Rainfall Data
Calculating river flow from rainfall - evaporation is a simplification. Firstly you are assuming that you can accurately measure rainfall, then that theoretical estimates of evaporation are correct which, given how much land cover and vegetation vary, is unlikely.
Next consider the pathway from raindrop hitting the soil to the river. It's clearly not instantaneous. That raindrop can evaporate directly, run off over the ground surface or soak into soil. In the soil it can be taken up by vegetation or infiltrate and become groundwater recharge. Depending on soil and aquifer characteristics this introduces a delay in the conversion of rainfall to runoff. In the real world this delay could be weeks, months or even years. In your example it's also worth thinking about low temperatures. Is rain falling as rain or is it snow?
Thirdly consider how water can get diverted from its pathway to a river. Is water abstracted from the pathway (i.e. from an aquifer),  or from the river directly?  Conversely reservoirs may regulate flows, either up or down depending on management.