Yes, you can always continue your progress.
On android, the newer .apk updates the game files automatically.
On all other systems, if you download the full game files again, you need to copy&paste the entire content of the www/save folder from the old game location to the new game location. Make sure you copy everything from the save folder, including the .global, or else the game won't recognize the save files when you start it and the "load" menu option will be greyed out.