Mealy machine (by G. Mealy) is a deterministic finite state transducer with output associated with transition (edge) instead of state. It could handle multiple inputs and multiple outputs but by definition cannot be non-deterministic. According to definition it must be defined for all possible combinations of states and inputs.
Finite State Transducer is a generalization of Finite State Machine (M. Rabin, D. Scott) which can be both deterministic and non-deterministic. The output is associated with states, not all combinations of states and input must be present. Not all NFST can be determinized.
The computational power of both is equal in the Chomsky hierarchy, but there exist machines that cannot be expressed by other type. In special transduction hierarchy FST is higher than Mealy.
There are extensions to create nondeterministic Mealy machines, but nondeterminism for transducers is not really exploited, it would generate the family of outputs when one is really needed, unless it was the goal then NFST wins here.
From physical point of view it makes difference, Mealy equivalent machines are shifted in time, but considering only the result it does not matter.