There is differences on many fronts.
Aspect ratio is probably the most noticable. A monitor with 4:3 aspect ratio would have a 25% smaller field of view as a 16:9 monitor so the difference is significant. 16:9 and 16:10 is negligible.
Someone with a higher resolution, thus bigger would have textures and models being represented by more pixels. This is insignificant in other games but with shooters, long range targets will be represented by fewer or more pixels. Running somethinkg like 800x600 might have a target look like 10 pixels you can only aim at one particular pixel and even one pixel might be a huge margin.
A screen at 2560x1600 would have 4x the pixels of a 1280x1024 screen. So targets are effectively 4x bigger and have 4x more surface area to aim on. Think of having a 4x scope on at all times but this scope doesn't limit your field of vision.
Things at higher resolution would of course look sharper and bigger as well with more detail but for gameplay it won't matter.
Larger resolution settings would tax only GPU performance. You need a bigger monitor and possibly a beefier graphics card to handle it. Larger graphics RAM is most important for high resolutions. 1GB is usually good for resolutions up to 2560x1600.