A pixel, imagine if you will, is essentially a "dot" on the screen. 1024x768 has 1024 "dots" across the screen by 768 "dots", or pixels, if we start using correct terminology vertically. Thus, at a given moment in time, the computer must display 786,432 pixels. Now, if you increase the resolution to 1440x900, the computer will have to generate 1,296,000 pixels at any moment in time. That's not a particularly large difference at all, and your computer, assuming it isn't like a P3 integrated solution, should have no problem running it.
Now, to answer your question, yes, higher resolution equals higher image quality. Ever seen HD tv versus SD tv compared side by side? Or a low pixel (2 megapixel) picture compared to a 10 megapixel shot? More pixels allows more detail to be drawn in.
The biggest difference you'll see in moving to a 19in is that you'll stop giving yourself eye strain.
15"? Man, how'd you last for so long?