**Yes, the two models are physically different.** The radio is different.

As [per Apple's website](http://www.apple.com/ipad/specs/), the radio frequencies aren't the same. The verizon iPad supports Verizon LTE, and as per [this post](http://www.macrumors.com/2012/03/16/new-verizon-ipad-runs-on-att-3g-networks-with-att-micro-sim/), AT&T 3G. However, *the Verizon iPad does not support all GSM frequencies that the AT&T iPad does*.

From Apple.com:

> Wi-Fi + 4G for AT&T model: 4G LTE (700, 2100 MHz); UMTS/HSPA/HSPA+/DC-HSDPA (850, 900, 1900, 2100 MHz); GSM/EDGE (850, 900, 1800, 1900 MHz)

> Wi-Fi + 4G for Verizon model: 4G LTE (700 MHz); CDMA EV-DO Rev. A (800, 1900 MHz); UMTS/HSPA/HSPA+/DC-HSDPA (850, 900, 1900, 2100 MHz); GSM/EDGE (850, 900, 1800, 1900 MHz)

Notice that there are some differences between the listed frequencies. They're nearly the same, but not quite.