What is the difference between Network Failover and Fail-safe on a LTM?

Network failover is a method used for certain HA communication between BIG-IP systems that are in a sync-failover device group. The alternative to network failover is to use a failover cable (hardwired failover), which physically connects the failover ports on two (and only two) BIG-IP appliances (not VIPRION or VE) in what's commonly referred to as a redundant pair. The "heartbeat" that is used to detect a failure is transmitted via a network signal for network failover, and via a voltage signal for the failover cable.

VLAN and gateway failsafe are configuration options that can be used to trigger a failover under certain circumstances. (They expand BIG-IP's HA capabilities.) For example, if the BIG-IP system detects no traffic for a specified particular period of time on a VLAN that is configured with VLAN failsafe, it may reboot, triggering a failover to the standby device.

Network failover is necessary if you have more than two devices in your sync-failover device group, or in the case of VIPRION and BIG-IP VE systems. You can use hardwired failover alone (or in tandem with network failover) on two BIG-IP appliances that are in close physical proximity to one another as there is a limit on the length of the failover cable. See SOL2397 for more information. (https://support.f5.com/kb/en-us/solutions/public/2000/300/sol2397.html)

With respect to whether or not to use VLAN or gateway failsafe, that depends a lot on your environment. See SOL13297 for information on VLAN failsafe (https://support.f5.com/kb/en-us/solutions/public/13000/200/sol13297.html) and SOL15367 for information on gateway failsafe (https://support.f5.com/kb/en-us/solutions/public/15000/300/sol15367.html).