Considering LTE is a direct upgrade for ATT (since it's the next generation of GSM) and verizon uses CDMA, I would say ATT will have far less problems. Verizon basically uses a bridging technology called eHRPD to hand off connections between LTE (GSMv4) and EVDO (CDMA). Their network problems with LTE were from a failure in eHRPD. Since ATT wont need this, they wont have that problem. LTE handoffs are a bit laggy on Verizon as well from my experience with having the HTC Thunderbolt since March. Generally when you switch from a CDMA area to LTE, it takes 10-20 seconds for the handoff to occur.
From wiki:
3GPP Long Term Evolution (LTE), is the latest standard in the mobile network technology tree that produced the GSM/EDGE and UMTS/HSPA network technologies.[1][2] It is a project of the 3rd Generation Partnership Project (3GPP), operating under a name trademarked by one of the associations within the partnership, the European Telecommunications Standards Institute.
http://en.wikipedia.org/wiki/3GPP_Long_Term_Evolution
http://en.wikipedia.org/wiki/LTE_Advanced (what will come after initial LTE we have now and be the true 4g before standards revisions)
http://en.wikipedia.org/wiki/3GPP
http://en.wikipedia.org/wiki/Ultra_Mobile_Broadband (att's current network)
article on what eHRPD is
Verizon has decided to use eHRPD as their upgrade path to 4G, which allows them to update their existing HRPD packet core using SAE/EPC architecture. The primary benefit that eHRPD offers is the handoff between cellular towers - you maintain the same private IP when you move from location to location. With this new protocol operators will be able to optimize cellular handovers, which should reduce dropped sessions and decrease the handover latency.