The difference between String Field Theory and String theory is, in the broadest of sense, the same as the difference between Quantum Field Theory and Quantum Mechanics.
QFT supercedes Quantum Mechanics in that it allows particle creation and annihilation dynamically. QM's basic building block is the Schrodinger equation, you're finding how a particle that you suppose already exists evolves when inside a potential that already exists, and never worry about how both of those things are generated. You can infer from that a whole lot more, but you will run into difficulties describing more complicated problems. In QFT, these things are not always fundamental, it allows us to derive a more microscopic view of e.g. the electromagnetic potential of a charged object, you can derive the famous 1/r^2 law.
String Theory is much the same, its basic building block is the action for one string as it moves through space. Again, from knowing that you can infer a whole load of stuff (particle states, anomalies, dimensions of space, existence of branes, etc.) but centrally String theory does not have a proper way of treating strings being created and annihilated dynamically out of the vacuum. We assume they can, and we can figure out many properties of what happens when it does, but the formalism is somewhat lacking.
String Field theory exists to re-employ the techniques that made QFT an improvement over QM in this circumstance to solve what seems like a similar obstacle.
In quantum field theory, you take spacetime as the domain in which all of the things you're studying (electron fields, electromagnetic fields etc) "live".
In **string theory** this is turned on its head, because you take the *string worldsheet itself* to be the domain on which all the things you're studying live. In particular, the spacetime coordinates that tell you the string's position are themselves treated as quantum fields that live on the worldsheet.
**String field theory** is the attempt to put this back the right way up: to treat spacetime as the domain and the strings as the things that live in that domain, instead of vice-versa.
For practical calculations such as scattering amplitudes, they give the same answers where applicable - but the way they arrive at those answers is fundamentally from a completely different direction. String field theory is still in a much less mature state than string theory - basic things from string theory such as superstring loop amplitudes are not well understood at all.
I hesitate to give such a short answer, but here we go.
> String field theory (SFT) is a formalism in string theory in which the dynamics of relativistic strings is reformulated in the language of quantum field theory.
I appreciate this summary because it implies the heart of SFT.
It is a "reformulation" of String Theory to match Quantum Field Theory (QFT).
In this sense, SFT is essentially a boiled down version of String Theory in order to replicate QFT.
One way of interpreting this ability of String Theory is that QFT is a subsidiary of String Theory (i.e. a person can derive QFT from String Theory if they interpret SFT correctly).
Now to your question.
String Theory is the over-arching theoretical framework that attempts to unify physics.
String Field Theory is a "simplified" version that attempts to mimick our current best approximation, showing that String Theory would be a parent of QFT (in some sense).
Quantum Field Theory is our current best approximation which we think is an approximation because of certain details that are left out (which is why we are researching String Theory).
This is my translation of the Wikipedia article.
A more detailed explanation would require someone with more expertise than I have.
Edit: I would encourage everyone to also ready u/rumnscurvy's comment. It has a good analogy that sheds more light on the relation between String Theory and String Field Theory than my clumsy use of quotation marks around simplified.
[Wikipedia Article on String Field Theory](https://en.wikipedia.org/wiki/String_field_theory)
At the risk of excessive simplification:
String theory is not a QFT, for a number of reasons. One key reason is that it is quantum gravity, and quantum gravity cannot be a QFT.
String field theory is an approximation to string theory which emphasizes the more QFT-like aspects of string theory in some regimes in which the theory is indeed QFT-like.
Recall that each vibrational state of a string is a particle. What you do in SFT is not much far away from introducing a quantum field-like object for each such particle-like object to build a QFT-like system.