Screen resolution is the number of pixels packed in a display. The higher the resolution, the sharper and more attractive images and text appear. However, this visual luxury comes at a cost.
Most affordable phones have HD or Full HD resolution. A typical HD (720P) resolution display has 921 thousand pixels while Full HD (1080P) jumps to a whopping 2 million pixels. These pixels have to refresh over 50 times every second that the screen is on (50Hz) ; this task is left to the processor. The more the pixels available, the more the work load. Therefore, screens with higher resolution demand more from the processor leaving the user with less power for apps and gaming. Hence, using the same processor, HD screens will perform better in gaming and general usage than Full HD devices.
The same argument applies. The more the resolution, the more the workload for the processor and in turn, more energy is drawn from the battery. When you are gaming, the fast scenes associated with games leads to maxing-out the refresh rate (60Hz) consistently. This overworks the processor and produces heat. Heat is a catalyst that quickens the process of discharging a battery. Therefore, using the same battery capacity, the lower the resolution, the longer the battery life will be.