What does QFT "get right" that QM "gets wrong"? Or, why is QFT "better" than QM?
There may be many answers.
For one example of an answer to a parallel question, GR is better than Newtonian gravity (NG) because it gets the perihelion advance of Mercury right.
You could also say that GR predicts a better black hole than NG, but that's a harder sale.
For QFT versus QM, I've heard of the Lamb shift, but what else makes QFT superior?
 A: Any quantum system at relativistic speeds is described in the framework of QFT. Or I should say: quantum systems at high energies or short distances (sub-nuclear) or short-time processes (sub-micro second stuff). For example relativistic Compton scattering is described by the Klein-Nishina formula which particle physics students derive in their QED course.
A: QFT is just a more powerful version of wave function QM. 
In statistical mechanics, it allows one to do many computations that would be awkward in terms of wave functions. 
In relativistic theories, it allows one to handle correctly the multiparticle situation which, for more than 2 particles, is extremely awkward to do without fields.
Also, it provides the connection between spin and statistics and the CPT theorem, 
which have to put in by hand in wave function QM.
Finally, QED and the standard model (with all their predictions) cannot be formulated without QFT.
A: The most important thing is that nonrelativistic QM (as it is formulated traditionally) cannot deal with changing particle number, because the position basis Hilbert space changes dimension as you increase the particle number. Quantum field theory allows particle number to change, and this is the main difference.
This is also important in cases where the number of particles is indefinite, like the fixed phase description of a BEC or a superfluid. In this case, the state of fixed phase macroscopic matter wave is a superposition of different numbers of particles. For this reason, nonrelativistic Schrodinger fields are useful in condensed matter physics, even in cases where the particle number is technically conserved, because the states one is interested in are in a classical limit where it is better to assume that the particle number is indefinite. This is exactly analogous to introducing a chemical potential and pretending that the particle number can fluctuate in statistical mechanics, in those cases where the particle number is exactly fixed. Or introducing a temperature in those cases where the energy is exactly fixed. It is still mathematically convenient to do so, and it does no harm in the thermodynamic limit. So it is convenient to use quantum fields to describe nonrelativistic situations where the particle number is fixed, but the behavior is best described by a classical collective wave motion.
