Why invariance is important? The concept of invariance seems to have a great importance. Indeed, the fact that the laws of Electrodynamics are not invariant in every inertial reference frame led to the theory of Special Relativity which in the end makes those laws be invariant.
As I understand invariance, a physical law is invariant in two frames of reference when it holds good in both of them. That means if we write the law mathematically, the law assumes the same form in both frames of reference.
So for example, Newton's second law is invariant in frames $S$ and $S'$ if whenever $\mathbf{F}$ and $\mathbf{a}$ are force and acceleration as understood by the viewpoint of $S$ and $\mathbf{F}'$ and $\mathbf{a}'$ are force and acceleration as understood by the viewpoint of $S'$ we have $\mathbf{F} = m\mathbf{a}$ if and only if $\mathbf{F}' = m\mathbf{a}'$, and the law is the same in both of them.
Now, since this idea of invariance led to something as important as Special Relativity and even led people to change the way they understand space and time I wonder invariance is a quite important thing.
So, is invariance really that important? If so, why do we care so much with it? What's the real importance of invariance?
 A: A theory is only useful if it can be applied to obtain predictions.
Lets consider you example in more detail. If galilean transformations hold, as they do in newtonian mechanics, then classical electromagnetism doesn't hold in all inertial reference frames. But nothing in the theory of classical electromagnetism holds one reference frame over another. Yet classical electromagnetism doesn't hold in all inertial reference frames. How can we decide then, which reference frame to apply the laws of classical electromagnetism? We cannot decide, rendering classical electromagnetism invalid.
Yet classical electromagnetism makes predicitions which seem to agree with expirements. Physicists came up with the theory of aether, wherein the laws of classical electromagnetism hold within the reference frame of aether, the medium of light. The only problem is that the experiments of Michael Morsely disproved the existence of a medium of light. In other words, there couldn't be a special reference frame wherein classical electromagnetism works. 
Considering that different expirements of classical electromagnetism occured across diffeent reference frames, classicla electromagnetism had to be seen as invariant across inertial reference frames, because the evidence supported this proposition. Yet this contradicted the galilean transformations of newtonian mechanics. Both theories couldn't be right at the same time!
Since efforts to introduce a special reference frame for classical electromagnetism failed, physicists decided to try to introduce new transformations to keep classical electromagnetism invariant accross different reference frames, changing newtonian mechanics in the process.
Many physicists including Voigt, Lorentz, and Poincare developed transformations to keep classical electromagnetism correct. Here is Voigt's derivation, which he did in 1887, if you view it on chrome you can translate the text from German. Einstein decided to focus on keeping the speed of light constant, resulting in a different, novel derivation of the Lorentz transformation. Here is Einstein's paper on special relativity, published in 1905, translated to English.
An excellent source is the Feynman lectures volume 2 section 26, on the Lorentz transformation.
In brief, invariance is essential to keep theories valid and useful, as well as to keep phsyics coherent.
A: Lorentz boosts are essentially rotations in the t-x plane (hyperbolic rotations, actually, or skews, but stick with the analogy for now), so it's often useful to get an intuitive feel for them in special relativity by comparing boosts to rotations on some other plane, like the x-y plane. So let's do that.
Consider if you were measuring the y-length of a stick on the x-y plane -- clearly, this depends on your frame of reference. A co-ordinate system in which the stick lies on the y-axis clearly gives you the maximum value of this y-length, a co-ordinate system in which it lies on the x-axis clearly gives you a value of 0.

So the specific co-ordinate dimensions $(x, y)$ of the stick depend on your reference frame. But we can also be interested in the real lengths of sticks, because this is invariant in all reference frames. This can be calculated easily using the Pythagorean theorem:
$$\psi=\sqrt{x^2+y^2}$$
(Note that the invariance is not the only thing that is important, but also that it allows you to define a polar co-ordinate system where $x=\psi\cos\theta$, $y=\psi\sin\theta$.)
If you accept that it can be useful to know the dimensions of objects on their own axes, it's clear that the same principle applies on the t-x plane. Here, the "rotations" are skews, the trigonometry is hyperbolic trigonometry, the Pythagoras theorem is $\tau=\sqrt{t^2-x^2}$ and instead of the proper time being the highest point of a circle it is the lowest point of a hyperbola.
But the same principles still apply -- if you see someone blast a toddler off into outer space at a high speed then return, you might measure the toddler as having taken a hundred years to return, but you and the toddler both agree (assuming he isn't dead yet from starvation) that he's only aged a year. This biological time, or proper time, is an invariant.
A: Invariance may be connected with two things:
First, invariance usually means a symmetry of the system under certain transformations. This symmetry and it's accompanied conserved quantities mean that there is a set of observational objects, that do not change. Moving epistemologically on this, we may deduce that the only way we can write down laws for nature, meaning for example equations of motion, is by having something conserved, since otherwise we would not be able to observe them.
On a second hand, invariance usually implies the non-existence of a certain notion, something impossible to observe, as absolute spacetime position. Thus, from this perspective, invariance and symmetry mean a way of distinguishing between observational and non0observational quantities.
