From a video perspective you will not notice much difference between a 1080p video being played back on a 1366x768 resolution and a 1080p resolution screen unless you compare them both side by side.
On a 1366x768 screen you will likely notice a little less detail (because lower resolution means you loose detail) and you might notice very, very slight image blur compared to a 1080p screen. This is due to something called video interpolation when trying to fit higher resolution video on a lower resolution screen plus the fact that you loose some detail as previously mentioned.
When not comparing how movies look on a 1366x768 resolution screen to a 1080p screen, then you will likely not notice any difference at all. Many people have enjoyed watching 1080p movies an Android tablets with 1280x800 resolution screens.
The issue regarding choppiness when watching 1080p video on your current laptop with the Intel GMA graphics is because the GMA generation graphic cores were not powerful enough for smooth playback. That has since changed. Current generation Intel HD graphic cores are capable of playing back 4k resolution video or Ultra HD. That's 3840x2160 resolution.
Naturally when playing games, the higher the resolution, the lower the performance. Therefore, you can simply opt for the 768p screen or just go with the 1080p screen and then drop the resolution down to 900p or 768p when playing games for better performance. Graphics will not look as sharp when using lower than max resolution because of the previously mentioned video interpolation. Generally, speaking the larger the difference between max resolution and the resolution you play games with the less sharp the graphics can look.
Having said the above, PS3 and XBox 360 games are all 1280x720 or 1366x768 resolution and stretched to fit on a 1080p HDTV screen. You generally do not hear people complaining how bad the graphics look because the resolution is stretched to fill a higher resolution screen. One the other PS4 and XBOne games should all be 1080p resolution.
Personally, I find 1366x768 resolution too limiting since I do more than just play games on my current laptop (Lenovo IdeaPad Y470). My next laptop will have a 1600x900 or 1080p (more likely) resolution screen.