The major difference between a new construction and retrofit window is the way the new window frame is attached to the house.
To install a new construction window the installer will have to chip away the original stucco, cut away some interior drywall, remove the window, remove the paper backing behind the stucco, and then once the new window is in ensure the window is properly flashed and re-wrapped properly to ensure everything is watertight. After that, they will patch the stucco which was removed to perform the window install.
In a retrofit install, the windows include a Z-bar or Flush fin which which allows the installer to cut down and reuse the existing window frame to mount the retrofit window frame to. The flush fin (Z-bar) covers the gap between the inside of the existing frame and the exterior of the replacement window and allows you to simply install the window from outside rather than tearing out the exterior stucco and interior drywall.
If your original windows were installed correctly and aren't leaking the retrofit windows would be the way to go in my opinion. You will lose a tiny bit of window viewing area using the retrofits, but the install time will be much faster (and of course much less expensive) and there's no worry about tearing up the inside and outside of your house. I've never seen patched stucco that looks totally right either. The energy savings will be there regardless of the installation method you choose.