Xbox 360 Slim unable to connect or find a wifi signal.

Im pretty sure that you cant use the old wifi adaptors with the xbox 360 slim because the internal wifi connection utilizes wifi-N, the newer standard of wireless connections and may not properly interface with the xbox 360 slim.

As for getting a better signal.... i dont think it really makes a difference, most issues regarding that usually are environmental issues surrounding the 360 and nearby access point or the router. For instance having thick walls, microwaves, other computers
running wireless, etc.

What specifically would allow the Xbox 360 Wireless Networking Adapter to connect while the built-in wifi would not?

My Xbox 360 S detects a wireless signal with 4 bars, and I am testing it while sitting it right next to the modem, so there is no issue with signal strength.

My Xbox 360 S's built-in wifi is able to connect to a hotspot, and it is even able to connect to my neighbor's wifi from my house.

Probably no, the new Xbox 360 slims have that same adapter built into it, if your having signal issues, you may be too far away from your WiFi access point, or the router's WiFi antenna isn't strong enough to maintain signal. it also could be that you have
a defunct WiFi adapter in you xbox. you can try the detachable adapter, but i dont know if it will conflict with the built in one. my suggestion is always a hard wire connection, it's faster and has more bandwidth then 802.11N will ever have. use a Cat5E or
Cat6 Ethernet cable if you can+^_^+

I don't know why it got not find you WiFi network but even when your Wifi is hidden you got find it normaly. But you need to do it manually like DrBogBrush said. I have a problem for myself at the moment that my Xbox 360 Slim will not connect with my WiFi
if my MAC address is enabled on the router. If i put this off the Xbox will connect right away. Maybe this helps you a bit.

I have the same problem with two airport express routers. My 360 connects to the router with the weaker signal and drops the connection. I have a mac mini sitting next to the 360 and it knows how to handle wifi networks like a mature adult and connects to
the stronger signal. My cheap hack is simply sharing the wifi connection of the mini over it's ethernet port and plugging the 360 into the mini via ethernet.

Very annoying, I have to be running my mac mini in order to have xbox live because the 360's wifi doesn't know how to connect to the stronger signaled network. This needs to be fixed. Windows machines seem to figure it out, why not a 360?