Well having a 1366x768 screen is about only 51% of the 1920x1080 resolution. So you'd have a huge different if your looking at pixels. Ideally I'd get the 1080P monitor to be a bit future proof even though 2k & 4k monitor's are almost affordable for the average Joe!
Obviously if your pushing video games on a smaller resolution you'll get higher FPS (Usually), but knowing most computers and customers configurations 1080P is more of a standard.
A little insight: I owned a HP W1358 Monitor it had 1366x768, mind you was amazing at the time (7years ago) I used to think it was the latest and greatest until full HD 1080p came out. Overall, my Netflix shows & digital movie library never looked so good!
Regardless if your thinking about upgrading, I'd say without a doubt get your hands on at least a 1080p (really cheap nowadays)
Physical size of the monitor measured corner to corner, resolution is the amount of pixels on the screen. pc monitors use 1080p starting at 19", 1366x768 is common in laptops and old/tiny pc monitors.
1366x768 (1049088 pixels) / 1920x1080 (2073600 pixels). Depending on the task, performance will be affected. As you can see 1080p is almost twice as many pixels as 768p, using your desktop at 1080p won't impact your pc performance in a noticeable way. Games on the other side will require more processing power.