I gamed primarily on my m11x for a few years, the 11.6" 1366x768 display was a meh TN panel with pretty bad contrast though. Not a problem with old games that used fewer colors, but newer games with shadow and lighting effects really don't look as great as on better panels.
Resolution is effectively resolving power, or ability to distinguish 1 object from another. Lower resolutions = more aliasing, fewer pixels. The usefulness of resolution depends primarily on the audiences eyesight, distance to the display, and size of the display. For example, lower resolution might make it harder to distinguish an object from the environment, requiring the player character to move closer in order to see some details, and also result in lines appearing jaggier.
I for the most part don't use anti-aliasing and resolution doesn't seem to be that big a factor for me, but YMMV. I can see the jaggies, it just never bothers me. I don't like screen-tearing, so I usually enable V-sync.
Playing games on a 50" 720p plasma >> 27" 1080p IPS HP 27xi >= 22" 1080p IPS HP ZR22w >= 22" 1080p MVA HP Elite L2201x >> 11.6" 720p TN
note: placement of the 3 desktop monitors is difficult, since I haven't directly compared them against each other, and time probably played tricks on my brain.
I prefer panel quality for better lighting/dynamic range/contrast compared to resolution
Most laptops on 1366x768 use TN panels for what it's worth (kind of annoying, since a lot of cheap 1366x768 tablets use IPS). Your 1080p might be TN or IPS. There are some good TN panels, but there are also some bad ones. Maybe look at notebookcheck for the notebooks in question, since they usually have pretty detailed breakdowns on them, though most reviews are for versions with the high resolution screen...
m11x display review for reference
http://www.anandtech.com/show/3620/alienware-m11x-worlds-smallest-gaming-laptop/7
this laptop came out before cheapo laptop displays started improving, likely due to the smartphone panel craze.
most of my qualms about low resolution have nothing to do with gaming.
It's rather annoying using 1366x768 with general web browsing and office work.