Yes, everything should look sharper larger and you will have much more screen estate to use.
EDIT: sorry I misunderstood you question. By smoother do you mean more frames per second? The lower resolution monitor will be less strain on your graphics and will probably be faster. So going to 1080p you will definitely need a more powerful pc to drive your games. It also depends on what games you play, your pc should be able to handle 1080p but in 3d games the graphics card part of your PC will have to work harder to render the extra pixels that go on the higher resolution monitor. If your PC can handle it, it will be well worth the upgrade.
For internet browsing the new space will actually be a huge improvment.
It is really hard to browse in 1024x768 as many sites simply hang out on the left as they are optimised for a minimum of 1200 horizontal pixels instead of 1024. I have a 19" 1024x768 monitor also as a secondary monitor at home, next to my 27" 2560x1440 montor. The 1024 monitor is a pain to use when searching on google. I only keep it because it really helps to have a 2 monitor setup. For example you search for somthing on google and put it on the second monitor while you actually do what you need to do on the main monitor. But still the 1024 monitor is a real pain in the butt when it comes to scaling webpages these days. Even 1080p is on its way out, so yeah it makes a huge difference in everything not just games.
Gabor