Your numbers are off, I doubt there are any 1020x1080 screens out there. That is a nearly square screen, in fact you have a width smaller than the height. (Assuming you are listing the WxH, as you have with the other, 1600x900 resolution).
You may have meant 1920x1080.
I am also going to assume that these are both the native resolutions of either laptop. (Native is where that is the fixed numbers of the display pixels). If you are considering using the Windows adjustment to change one to the other, I don't recommend that. For one thing adjusting the screen resolution, via some control, may not be allowed within the gaming program. And even if you can change the display resolution, any resolution not the native resolution will result in a less than perfectly clear image. At least, not as clear as can be done with that screen.
The aspect ratio of both (1920x1080, and 1600x900) are the same, at 16:9.
The number of pixels to move around the higher resolution (1920x1080) is about 33% more than the number of pixels on the 1600x900. Having more pixels to "push around" can slow down some display rates. But that also depends upon the hardware and the drivers. Since you are saying this is going to be a gaming laptop, you may care about how fast the screen display is.
Being a laptop, you will have little chance to upgrade the video hardware (not impossible, depends upon the manufacturer... read that as can be very expensive). You will have times where you can upgrade the video driver, and while most video updates usually are made to improve the video performance (usually), but sometimes an updated driver might slow the performance down too. That might happen to resolve some reliability issues.
I'd say, if you like the lower resolution 1600x900 display, go with that.