Since you are getting a valid IP address for the wired connection, your connection problem there is probably related to DNS. The router should be giving you the address(es) for DNS servers, but if not, you may have to enter them manually. Check the machine that is working to get the IP address(es) for DNS servers. As mentioned, wireless security has nothing to do with getting the wired connection working.
As for the wireless connection, as I've said in these forums many times before, I hate wireless security. It is confusing and causes more headaches with debugging connection problems than anything else. You say you can connect, meaning that you are able to enter the password. Do you get a valid IP address? You say you do for the wired connection, but not for the wireless connection. If you can test with wireless security turned off, then you can get that out of the equation of why you are having a problem.
Also, if the router is set up to allow connections only from certain machines using MAC address filtering, your new machine may not be in the "whitelist" to allow a connection. Basically, you need to get access to this router to check the settings on it to allow your machine to connect.
Steve