Do same size screens with different PPIs affect battery differently?
Solution 1:
Between two LCD displays identical aside from the physical PPI (and resolution), the one with the higher PPI (and resolution) will use more power for at least two (related) reasons:
- More pixels = more transistors to power
- More transistors on the panel = reduced light throughput = more power needed to match the brightness of the other display.
Sending a non-native resolution signal to the monitor may actually increase power usage (of the display) by necessitating the use of a scaler circuit that would otherwise be off, or may decrease it by reduced need of other resources. Note that this does not account for a change in power usage outside of the display (e.g. GPU, which may increase or decrease based on a variety of factors).
The day you chose to ask this question happens to be pretty fantastic. Anandtech just published an article comparing the QHD+ and 1080p display options for the 2015 Razor Blade 14".
It's not perfect, as there are more differences between the displays than the PPI/Resolution and they are counting the entire system power, not just that of the display. But they found 17-22% longer battery life with the 1080p than with the QHD+.
Solution 2:
No, the resolution of the screen will not affect battery life. The reason for that is that, as you said, the actual number of pixels is the same. The monitor cannot turn pixels on or off, as this would literally cause there to be little black spots visible on the monitor given that the pixel size is hardware based, and cannot be changed on the fly.
UPDATE: If PPI is different but everything else is the same then the number of the pixels on the monitor is different and thus the power consumption will be different (the one with higher PPI will use more power).
The below is a more detailed explanation by analogy, to help give you an idea of how monitors actually work to virtually "scale" pixels. Note that the monitor still uses all available pixels, regardless of what it is displaying.
Zooming in too far
Think of how an image is scaled up or down on a computer screen. When you zoom too far in, it gets blurry, because the size of each pixel in the image begins to take up more than one pixel on the monitor.
To display the image, the monitor's pixel size actually does not change, but rather it begins to use multiple monitor pixels to display a single image pixel.
This means that the image has gained additional density, however the monitor has not gained or lost any density and is still displaying the same number of pixels.
Zooming out too far
To make the issue more clear, try to think about that in reverse. When you scale an image down too far, the monitor's pixels become larger than the the pixels in the image.
Imagine you scale an image down to half its original height and width. For the monitor to properly display the image, each of its pixels must represent four of the image's pixels (two in each direction, making a 2x2 square). The monitor does this by having the pixel display the color calculated by taking the average of the four pixels on the image that it represents.
This means that the image has lost some density, because the monitor does not have enough pixels to display the image in full at its current size. We say that "the monitor does not have enough pixels" because, again, pixels cover the entire surface area of the screen, and the number of pixels is limited by the size of the pixels.
To clarify, it is equivalent to say that the monitor's pixels are too big to display the image in the fullest possible quality at its current size, because the image's pixel density at that size is greater than the monitor's. The monitor still displays the image in the highest possible resolution, however, and still uses all of its pixels to display the image.
More info
Have a look at this post about image quality and pixel density on Retina displays (2x HD quality of an ordinary HD display). It may help you understand how monitors will "scale" pixel sizes to match the content, while still using all available pixels on the screen.