Are there any physical differences between the Verizon and AT&T iPad (3rd generation)?
Yes, the two models are physically different. The radio is different.
As per Apple's website, the radio frequencies aren't the same. The verizon iPad supports Verizon LTE, and as per this post, AT&amp;T 3G. However, the Verizon iPad does not support all LTE frequencies that the AT&amp;T iPad does.
From Apple.com:

Wi-Fi + 4G for AT&amp;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.
Edit:
To confirm, the difference is a hardware difference, not a software imposed limitation on AT&amp;T LTE. From the Macrumors article:

The Verizon iPad hardware supports only the 700 MHz band for LTE to match the carrier's technology, while the GSM version sold for use on AT&amp;T's network supports both the 700 MHz and 2100 MHz bands as used by that carrier.

If you're looking to travel internationally, go with the AT&amp;T iPad. (I got the Verizon iPad because I wanted the free hotspot feature, which AT&amp;T does not offer for free at this time.)