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&T 3G. However, the Verizon iPad does not support all LTE 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.
Edit:
To confirm, the difference is a hardware difference, not a software imposed limitation on AT&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&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&T iPad. (I got the Verizon iPad because I wanted the free hotspot feature, which AT&T does not offer for free at this time.)
Best Answer
While I, personally, haven't had any experience using the iPad in a corporate setting, there are a couple people in my company who are using it in a lot of meetings and seem to be perfetly happy with it for lightweight tasks: taking notes in meetings, quick responses to emails, and the like. Your mileage may vary.
I'm pretty sure the iPad only has the wifi and 3G and I'm not aware of any LAN connectivity unless someone has sorted out an app for that (something like Apple's iDisk app.)
There are several remote access apps out there, LogMeIn has one that comes to mind but that costs a quite a bit of money. My friend uses that and really likes it a lot. but this is to log into a specific machine, not just jump onto network like a laptop would.