The difference between the three terms above is that vMotion™ is a trademarked name for a company product, while the other two terms are general terms referencing the methods of migrating virtual machines in a network.
With VM migration, IT engineers move virtual machines between physical machines without interaction. In hardware virtualization, a virtual machine or VM is a virtually constructed entity that works like a physical computer would within a network. It has its own memory and CPU, but these are allocated to it by the system, rather than being built in. But in many cases, it’s necessary to migrate these virtual machines, to send them from one part of a system to another, in order to deal with issues like workload balancing or CPU contention.
Likewise, 'live migration’ just refers to moving these virtualized machines around in a virtual environment. Strategies for live migration include accessing higher bandwidth for faster migration, configuration strategies and the use of clusters to achieve these goals. The key thing about live migration is that this term implies that everything is being done without interruption to the running servers.
By contrast, vMotion™ is a specific tool from the VMWare company. VMWare maintains that with vMotion™, engineers can migrate running virtual machines with zero downtime. This provides a branded solution for doing effective live migration, which necessitates reconfiguring a system without affecting its currently running parts.