WiFi connection to Xbox 360 with wireless adapter at hoteli

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.

the first version of the xbox 360 needed a wireless adapter, but the S and e models included wifi so those models should be able to connect via wifi just fine

And, if you can buy or borrow a wireless adapter, try connecting by WiFi if your router has WiFi. New Xbox branded wireless-N adapters go for $60

Do you have another wireless adapter that you can connect to your Xbox 360 Arcade (white console)? As Jonessyy mentioned, this could be a faulty wifi adapter and not a problem with the console.

The Xbox 360 internal wireless adapters do not support 5Ghz wifi bands, only 2.4Ghz.

The Black Xbox 360 external N wifi adapter supports 5Ghz.

5Ghz may support higher bandwidth (only helpful for LAN transfers and fiber/gigabit connections) but 2.4Ghz has more range.