Tight vs. overtwisted contact structure
As mentioned in the previous answer, the first use of the overtwisted/tight dichotomy is most certainly Bennequin's Theorem that there are non-isomorphic contact structures on $\mathbb{R}^3$ and $\mathbb{S}^3$, a landmark result.
However, the relevance of this dichotomy goes now far beyond this. As you probably know, contact topology has a Darboux theorem: locally, all contact structures are the same, isomorphic to the standard contact structure on $\mathbb{R}^3$. So, all of them are "locally tight", and having an overtwisted disc must be a global condition. A global way of distinguishing objects which are locally the same is tremendously important, here it can be thought of as some analogue of Gromov's non-squeezing theorem in symplectic geometry.
Moreover, while the classification of overtwisted structure has been achieved quite early by Eliashberg (Inventiones 1989), the tight contact structures happened to be very rich (see e.g. Giroux, Inventiones 1999 and its introduction -- in French) : certain manifold have only one of them, but infinitely many are shown to bear infinitely many non-isomorphic tight contact structure in the paper cited. The relation with symplectic fillings is one more indication of the relevance of this dichotomy. One can now consider that the tight contact structure are the one to study, as overtwisted ones are pretty well understood.
Note that the relevance of this dichotomy has recently been more firmly established in higher dimensions too by Niederkrüger, Massot and Wendl (Inventiones 2013).
About Bennequin's theorem, I can not explain its proof, although I learned one in my graduate years. I cannot give a complete account of it any more, but I can give some flavor. Consider the usual cylindrically symmetric version of the standard contact structure, and decompose $\mathbb{R}^3$ as a trivial open book (the vertical line through the origin is the binding, and each half plane it bounds is a page). Assume there is a closed horizontal curve(in the sense of the contact structure) which bounds a disc tangent to the structure, which you assume in general position, and look at the way this disc intersects the pages. In each page the non-transversal points are vertices of a graph, so you get combinatorial objects to work with. That's pretty much what I remember, but there are several proofs in the literature (at least one by Giroux, but likely to be written in French; it is probably not a waste of time to learn enough French to read mathematics if you are planning to work in contact topology). In fact, a friend of mine working in this area once told me that it was almost a duty for anyone working in this area to find its own proof of Bennequin's theorem. You should be ale to locate several proofs through the literature, but I am too far from this field to help you there.
Perhaps I'm wrong but I thought that historically the notions were considered first in the work of Bennequin who proved the existence of contact structures not equivalent to the standard contact structure on R^3 just by using that they have overtwisted disks and by showing (I think this was his main result) that the standard contact structure does not have overtwisted disks, i.e., is tight. see Seminaire Bourbaki
As Douglas Zare said above, the discovery of the dichotomy between tight and overtwisted contact structures is certainly inspired by the existence of taut and non-taut foliations in dimension three (or probably more correctly between Reebless foliations and foliations with a Reeb component).
It is not very hard to imagine that if you knew the turbulization procedure for foliations (that is, if you have a loop that is transverse to the foliation, then you can modify the foliation in a neighborhood of this loop to create a Reeb component), then you would think that the construction is quite similar to a Lutz twist (creating a loop-full of overtwisted disks close to a transverse loop in a contact manifold).
The properties of overtwisted contact structures and foliations with a Reeb component are relatively similar, for example, every 3-manifold admits them, but it's not true that every 3-manifold admits a tight contact structure or a Reebless foliation, by performing a Lutz twist or a turbulisation you can modify a contact structure into one that is overtwisted and a foliation into one that has a Reeb component, but it is usually very hard two undo these constructions. As you said in your post the behavior of submanifolds is much more restricted if the contact structure is not overtwisted/if the foliation is Reebless, etc.
The main difference between the theory of foliations and contact structures is probably that in the latter case you have Gray stability which tells you that contact structures are stable under continuous deformations.
Nonetheless note that by a recent result
Eynard-Bontemps, Hélène, On the connectedness of the space of codimension one foliations on a closed 3-manifold, Invent. Math. 204, No. 2, 605-670 (2016). ZBL1345.57033.
you can always find a path through foliations to connect any two foliations that are homotopic as plane fields. By contrast the paper
Bowden, Jonathan, Contact structures, deformations and taut foliations, Geom. Topol. 20, No. 2, 697-746 (2016). ZBL1338.53053.
shows that if you suppose that the two foliations are taut, it might be necessary to move through non-taut foliations to find the connecting path. Of course it is true that the last two results I cite are very recent, but comparing them to
Eliashberg, Y., Classification of overtwisted contact structures on 3-manifolds, Invent. Math. 98, No.3, 623-637 (1989). ZBL0684.57012.
should show that there really are many formal links between both theories.