TITLE: Tight vs. overtwisted contact structure
QUESTION [7 upvotes]: I know the familiar differences between tight and overtwisted contact structures. For example,  each homotopy class of plane-bundles on a three-manifold has an overtwisted representative but tight contact structures are less common. Also a tight contact structure gives a genus bound on an embedded closed surface and therefore can tell things about the topology of the ambient manifold. My question is what were the motivations behind distinguishing between the two originally? Is there possibly an intuition which makes the two fundamentally different and then proving the above differences comes as a consequence or it was these differences which led to distinguishing between the two? In other words I am trying to see the difference between the two without appealing to the statements like the above.

REPLY [4 votes]: 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.