|1) Why will it see the Netgear router, but not the Asus router?|
Did you configure the ASUS with the exact same SSID/password you used on the Netgear?
Also, you'll want to put them on separate channels. I suggest you put one on channel 1 and the other on 11
2) Could I connect both routers to the modem to alleviate the problem? If so. how
The best way to do this would be to connect one router to the other with a network cable. This is called "daisy chaining". For info on how to do this, click on my name above in my response and read my "how-to" guide titled, "Add a second Router to your LAN". You'll want to use the scenario where you connect them "LAN port to LAN port"
If you have to do it wirelessly, only one router should be put in bridge mode.
It matters not how straight the gate,
How charged with punishments the scroll,
I am the master of my fate;
I am the captain of my soul.