Why exactly are fields so crucial to modern physics? I have read that fields were the mathematical tool that allowed people like Laplace to develop a working model of the Solar System where Newton could not. But my understanding is that fields are nothing more than
$$E = \frac Fq$$ (where $q$ is either charge or mass depending on the field)
But doesn't this just simplify Newton's equation for gravity from
$$F= \frac {G*m_1*m_2}{r^2}$$ to $$F= \frac {G*m_1}{r^2}$$ and the equivalent for the electric field?
This doesn't seem like much of a simplification, so I must be missing some key concept. Why is it that the concept of fields allowed Laplace to do what Newton could not?
 A: Laplace's great achievement was the realization that the gravitational potential at position $\mathbf{x}$ due some mass $M$ at position $\mathbf{x'}$ can be expressed as
$$V(\mathbf{x})=\frac{M}{|\mathbf{x'}-\mathbf{x}|}$$
The gravitational field, $\mathbb{g}$, is related to $V$ by
$$\mathbf{g}=-\nabla V$$
where $\nabla$ is the del operator. The important part of Laplace's analysis wasn't related to $\mathbf{g}$ at all, but to $V$! He was thus able to work with spherical harmonics, which turned out to be extremely helpful. $V$, of course, is related to fields - the force field at each point in space, that we call gravity!

Something that should be made clear is that Newton's law of gravitation leads to a field: the vector field describing the force of gravity at each point. The same goes for electricity. The quantity we call the electric field is a field, but so is the force that arises from it, which we obtain from Coulomb's law.
Newton used fields, as did Laplace. Laplace simply worked with a different quantity.
A: The change from forces to fields isn't about simplifying the math. It's about changing the subject from the interaction between objects to a field and the source of that field. Prior to Faraday's concept of "lines of force", physicists would speak of one object's effect on another object some distance away. Famously, Newton explicitly offered no hypothesis as to what allowed distant objects to influence each others' motion. He just gave the effect the name "gravity." Faraday's conceptual breakthrough was to not speak of two objects influencing each other, but of a single object generating a field around itself and then having the field influence other particles.
To be more specific, the language changes from


*

*"A positively charged object exerts a repulsive force on another positively charged object."


to


*

*"A positively charged object generates an electric field in the space around it that points away from that object. If another positively charged object enters this field, it will experience a force in the same direction as the field."


As far as object interaction goes, all that changed is that there are a lot more words to describe "like-repels-like." However, there are two big gains from the field concept.
First, it removed the mysterious action-at-a-distance problem that gravity, electric, and magnetic theories had by having an entity of some kind reaching from one particle to another.
Second, and more importantly, it allowed the study of the field itself, apart from its effect on other object. It was discovered in field equations that, if you shake a charged particle, the electromagnetic field around it also shakes. This shaking moves down the field at a certain speed, which was calculated by Maxwell to be the speed of light. It was here that light was discovered to be oscillating electric and magnetic fields. I consider this to be one of the greatest wins for field theory, for without the concept of fields, the nature of light would not have been discovered. The two-object, force-centric understanding of electric, magnetic, and gravitational interactions doesn't include a concept of anything happening in between the particles, nothing that travels from one to the other. Indeed, half a century later, Einstein's field equations for gravity lead to the concept of gravitational waves--undulations in spacetime that also travel at the speed of light.
Fields mediate the interaction between distant objects, and that makes them worthy of study in their own right.
A: Definition: A field is any differential map $\phi\colon M\to N$ between two manifolds (or any other sort of space, at will)
Examples: 


*

*the path of a classical non-relativistic particle in one dimension, 
$x\colon \mathbb{R}\to\mathbb{R},\quad t\mapsto x(t)$

*the electric field at any point in space and time, $\mathbf{E}\colon \mathbb{R^4}\to\mathbb{R^3},\quad (\mathbf{r},t)\mapsto\mathbf{E}(\mathbf{r},t)$

*the wave function in quantum mechanics, $\psi\colon \mathbb{R^4}\to\mathbb{C},\quad (\mathbf{r},t)\mapsto\psi(\mathbf{r},t)=\langle\mathbf{r},t|\psi\rangle$


In physics most of the times things are described in terms of variations of such quantities (think at the Newton's equations of motion for the position, or the Maxwell's equations for the electromagnetic field, or the Schrödinger equation for the states in quantum mechanics) and therefore people have developed tools and machineries to understand how the above quantities behave. Even more, it is assumed that in order to derive the correct equations of motion, a general action principle is given in terms of the aforementioned fields, so that the true dynamics can be derived simply applying some mathematical tools (taking variations thereof).
In fact, given $\mathscr{F}(M,N)$ as the set of all such fields and $T\mathscr{F}(M,N)$ as its tangent bundle, one defines $S\colon\mathscr{F}\times T \mathscr{F}\to \mathbb{C}$ to be the action, $S\equiv S\left[\phi, \partial_{\mu}\phi\right]$. The equations for the dynamics are obtained implementing the condition
$$
\delta S\left[\phi, \partial_{\mu}\phi\right]|_{\phi=\phi^*}=0
$$
$\phi^*$ being the solution for the dynamics.

Why is it that electric fields allowed Laplace to do what Newton could not?

Field theory has very little to do with Laplace and Newton, who, by the way, did find a good model of the solar system (what more than the gravitational law would one want?). Much has been acquired with the work of Maxwell, who put down in decent and readable terms the equations for the electric and magnetic fields in terms of divergencies and curls of vector fields (or equivalently, integrals over surfaces and closed domains); ever since, people have started realising that it is (almost) all about the mathematical properties of such differential maps, together with their derivatives, and that general theories can be all derived, mutatis mutandis, using the same mechanisms and machineries (the standard model, for example, comes from field theory on gauge bundles and it perfectly works, having been experimentally proven).
