Why does hybridization significantly matter for organic chemistry?
Orbital hybridization is mostly useful in the rationalisation of molecular geometry. In fact it was invented by Linus Pauling for this purpose: to rationalise structures of organic molecules using the language of atomic orbitals. Yes, hybridization is also used to rationalise bonding properties, as well as acidity or basicity, but its primary use is to rationalise molecular geometries.


  I had the impression hybridization regulates a standard bond among
  same species of atoms, e.g $\ce{C-H}$ bond in all alkanes are the same. But
  that's not the case when I read brief comments indicating that the $\ce{C-H}$ bond in $\ce{CH4}$ differ from $\ce{CH3-CH3}$.


Well, $\ce{C-H}$ bonds, of course, differ in $\ce{CH4}$ and $\ce{CH3-CH3}$, but not too much; they are approximately the same. Are you naive enough to hope that a model as simple as the orbital hybridization one would perfectly predict the molecular geometry of each and every organic molecule? Surely, it won't. It can just give a crude (but simple) approximation.


  What I am unable to reason is, why hybridization is so important for
  organic chemistry than to any other (general, inorganic..., physical)?


Where did you get that? Hybridization is a concept of general chemistry, it is used in both organic and inorganic one. Its relatively big importance for organic chemistry is due to the fact that it is the only simple model which can explain (approximately) molecular geometry of organic compounds. The original valence bond theory couldn't do it, so it was extended.



On explanation vs rationalisation

Now I would like to emphasize that hybridization provides a rationalisation of a molecular geometry (as well as some other properties), but not an explanation, so it can not be used to reliable predict the properties. What is the key difference between an explanation and a rationalisation in these context? 

Well, an explanation of some phenomenon describes it in way that we can observe and test the consequences of our claims and premises. We can then tell from the observations are our claims and premises true or not, or, in other words, does our explanation work or not. The whole point of an explanation is to distinguish what is true from what is not true, in a sense, what corresponds to the actual reality from what does not do so.

In a rationalisation things work differently. We insist on some premises being true without testing them against the reality. We do observations, but we do not use them to tell are our premises true or not, rather on the basis of the desired premises and the observable evidence we start to make the observed outcome seem consistent with the premise.

In psychology rationalisation is also known as making excuses. It is a defence mechanism in which a controversial (from the society viewpoint) behaviour is justified "in a seemingly rational or logical manner to avoid the true explanation". With respect to scientific theories, it can also be thought of as a defence mechanism: when the real explanation is too complex for beginners we avoid it to defend them. Instead we first provide some simple rationalisation and later in the course of study the explanation is introduced.



The way rationalisation works for all these general chemistry models is as follows:


We do know geometry or some other property of a particular molecule, say, from an experiment, and we are seeking for some rationalisation of why the things are the way they are.
The rationalisation usually has to be as simple as possible, so that, for the case of molecular geometry, we usually start from VSEPR theory.
If the simplest possible theory can not rationalise the observed property, we choose a more sophisticated model. For molecular geometry, for instance, we could then try to use the original valence bond (VB) theory.
But even the VB theory (in its original formulation) won't work for organic compounds, so it has to be extended by two concepts: resonance and orbital hybridization. Note that this concepts are also required in inorganic chemistry as well, but not always: for some inorganic molecules the original VB theory is enough.


But however deep you go into this hierarchy of chemical models, all them are just rationalisation models: they provide simplified (and in many cases oversimplified) descriptions of the reality which is approximate (and in many cases rather crude). And besides all this process of seeking for an rationalisation is a bit like cheating:


If some model doesn't work, let us try another model.
If none of them works, let us introduce a new concept with some funky name in one of them, the purpose of which will be to rationalise this particular thing which couldn't be rationalised by the original model.


As a result, the predictive power of such models is quite small. They can be used to rationalise what is already known, but any predictions based on such models will be unreliable. And yeah, the concept of orbital hybridization is just one of these cheats mentioned above.