So Heisenberg ushered in the quantum revolution by introducing Hermitian matrices $x,p$ satisfying $xp-px = i\hbar$ and evolving according to the equations $$i\hbar ~\dot x = H x - x H,\\
i\hbar~\dot p = H p - p H.$$
Since the above commutation relation gives $p = -i\hbar \partial_x$ but also $x = i\hbar\partial_p,$ one can see an echo of Hamilton's equations in here. For Hamilton $x,p$ are coordinates for the phase space and $H$ assigns to every point on that space a number; for Heisenberg $x,p$ are numbers to be measured in an experiment and hence it was not clear at first glance what the phase space was meant to be replaced with, or what the matrices operated upon.
On the flip side, we knew that Planck and Einstein were driving at some abstract point about the quantization of energies of electromagnetic radiation, but that electromagnetic radiation was described with some fields $\vec E(\vec r, t), \vec B(\vec r, t)$. And we knew that this was not an isolated point about it: Einstein had based his theory of special relativity upon those same fields, and his theory demands that all information travel slower than $c$. (One reason why: two people passing each other will both say that the other person's clock is ticking slowly. If they had instantaneous information transfer they could in principle call each other on a telephone and just see, who is talking slower than the other one and settle which one is absolutely in motion. The time delay makes this question impossible to answer.)
Requiring that sort of influence to propagate at a certain speed or slower seems to demand this notion of a field between: you need to say "the effect of this motion has not gotten to where you are yet" which implies that you have a notion of where the force is and where it is not, which is a field description of that force.
In that sense it was only a matter of time before folks tried to use a field theory with quantum mechanics. But here your degrees of freedom in configuring the field are no longer $x, p$ -- in fact this is a step back from Heisenberg, those are back to being a "landscape" on which the theory lives. But your degrees of freedom are the infinite set of field values at all of the different points of space, $f(\vec r, t)$.
And so it is the basis of QFT that those numbers are promoted to matrices. In other words you invent an abstract Euclidean/Minkowski space, you have a scalar field assigning to every point in that space some scalar number, and we do quantum mechanics by leaving the space alone but promoting the scalar field to assign to every point in space a Hermitian operator.
Now the question is why that would be necessary once relativity comes into the picture, and the typical answer is the changes relativity seems to suggest in the quantum nature of particles. Traditional quantum mechanics is based on a model where a two-particle system promotes a 4-variable phase space $x_1,x_2,p_1,p_2$ into four matrices $x_{1,2}$, $p_{1,2}$. This derivation seems fine as long as the phase space stays the same. But relativity comes in and apparently indicates that we need to consider (if only by rotating Feynman diagrams in spacetime) that new particles might be created or destroyed in our physics. The phase space is changing! This translates to a great difficulty in defining the Hilbert space where the wavefunction lives.
That's where it's very nice to just have multiple fields living atop a shared physical space rather than independent physical spaces for every particle. As a bonus, we in some sense get "for free" that there can be two excitations in the "electron field" and that they represent totally identical particles, which quantum statistics was apparently demanding but which seemed absurd: you're looking at a proton coming in as a cosmic ray, it has been travelling over so many light years of space from some ancient event, and you are telling me that it's exactly the same as a proton I got out of the beta decay of a free neutron five seconds ago?! This is more plausible if they're both just "excitations in the same proton field" or so.
So the difference is that QM lends itself to combining configuration-spaces with a "tensor product" into a gargantuan mess and has difficulty changing its dimension; QFT starts out with an even more gargantuan mess, but gets away with never having to change it.
There are other slight differences -- technically QM is based on the Hamiltonian mechanics of a system while QFT is based on its Lagrangian mechanics -- but I think those might be comparatively minor.
