Keeping your programs up to date is always a good idea. Security vulnerabilities are unlikely to be exploited in OpenTTD but still, some exist. Also, new versions of OpenTTD include Bugfixes and new features, almost all people play on Multiplayer using latest stable version and using outdated versions is not recommended. Also online content like NewGRFs, Scripts should be updated.
It is a good idea to update on encountering some problems, it is likely that it is fixed in recent version (if bug is present also after update - please report it).
It is possible to automate updating by using OpenTTD Updaters.
To manually update on Windows - download installer from http://www.openttd.org/en/download-stable and run it.
Updating downloaded content
It can be done using Online content: open online content window as described at Online content#downloading-content, use the Select updates button in the lower left corner and click the Download button in the lower right corner.