Specialy "high end" usb-serial adapters often have "drivers" that include port mapping software. basically this interface tells the computer that USB port (name of the port) is mapped to serial port (port number). In finicky devices such as some medical equipment and most likely your weather station, the the numbers/names of the ports can be very important.
The sequence would be to remove all weather station software. Remove any existing port mappings. Reboot... A "cold" boot (all the way off and then back on).
Then install only the usb-serial adapter and software. Make sure it functions (try a mouse on it for example). If it works, do another cold boot.
Then install the weather station as per instructions.
If it does not work, then check the firewall settings... better yet, on a temporary or test basis, disable the firewall (Windows/CA/Norton) and make certain ALL firewalls are disabled. Often Windows update will re-enable the Windows firewall, or Norton will re-enable their firewall and you can easily have two running at the same time.
Then try the Weather station setup again.