Configuring the wireless part of the router is simple -- but I would urge you to download and read the WGT624 manual from www.netgear.com first.
Most likely the issue is with wireless encryption and this can easily be proved by turning it off temporarily in the router having connected the computer to the router via ethernet as you have been doing.
Then just try to figure out which security standards are supported by both computer and router -- most likely what the router will call WPA PSK TKIP (though Vista probably pointlessly calls it something else).
You aren't the first person to assume that you have to set up each connection to the router using the CD which came with the router.
In fact, the CD is merely a convenient tool to set up the router initially. Once the router has been set up the connection between the router via wireless is down Windows controlling the wireless adapter and you supplying the necessary security details (if security has been set in the router at all).
I suspect it worked because Windows eventually configured it correctly after you had re-started the computer -- as this will clear the IP address if it has gone astray. You can also do this manually via the Command Prompt by typing IPCONFIG /? and choosing the /all option to see what's wrong and the renew or release options to clear errors.
Personally I also always use the utilities that come with the wireless adapter to configure new connections as they are usually less opaque than the Windows equivalent (the downside is that they consume more memory than the Windows utility)
First step is usually to turn off wireless security in the router using another computer that is connected or to first connect the new computer by ethernet cable.
I would not use the CD which came with the router as you risk messing up the settings required to connect to your ISP. As I said, the router CD is intended for setting up the router, not wireless adapters.