On android, you download and install the newer .apk. It'll automatically recognize that you have the game already installed, and update it instead. Your saves are not changed in that process and can be continued after the update.
On all other systems, you have two options:
- You could download the full game again, and copy&paste the entire content of the www/save folder from the old game version to the same folder in the new game download.
- Or, if you have the previous game version and didn't skip an update, you could use the "update files only". These are only the files that have been changed since the last update. The version number for which version the update files are is always in the file name. It's possible to install multiple update files in order, but usually when you missed a few updates, it's better to download the full game again, to avoid errors for missing files when something went wrong.
I made a video about both options here: