According to the official website, the program for the weather stations was developed to run on Windows 98 which was a MS-DOS based operating system. This feature was changed in the newer versions of Windows, like Windows 7. Because of this, there are little chances that you will be able to use programs developed for Windows 98 on newer operating systems. This is the main reason why the weather station program will work on your new computer. To fix this problem, you can contact the developer using the site mentioned and ask for an update of the program, but probably there isn't one.
You can use a tool like VMware in order to emulate an older version of Windows on your new computer and run Heavy Weather program from there.