Thanks for your time.

When I run wpa_supplicant with the proper arguments it connects me to the Internet while within the Arch live CD, or when arch-chroot is run and I execute it from the emulated "hard disk" environment (after booting into the live CD first). In these cases, both the CD and the HDD installations recognize my wireless connection as "wlan0"

When rebooted without the live CD, not only does ip link show the wirless device as "wlp2s0" which I have never seen before, it also fails to connect (I cannot ping google) even though after running wpa_supplicant with the exact same arguments I get the message:

CTRL-EVENT-CONNECTED - Connection to <snipped> completed

Does anyone have any idea why this is?

Thanks again.

First off wlp2s0 is within the new naming convention that Arch Linux is using; wlan0 is the older (and more conventional) naming convention. They've changed as the new convention tells you more about the connection.
I'm not a massive fan, but I understand their reasoning.

I unfortunately cannot help, at least not directly.

You may wish to check the ArchWiki Wireless Setup.
It's a little out dated with some of the naming conventions. However, it should point you in the right direction.