I would like to understand what is the importance of large gauge transformations. I read that these gauge transformation cannot be deformed to the identity, but why should we care about that?
They require a special discussion because they are different. The (defining) fact that they can't be deformed to the identity means that it is not enough to verify the invariance under infinitesimal gauge transformations: the problem is that the large gauge transformations cannot be obtained by combining many infinitesimal gauge transformations!
The modular invariance of the torus in one-loop diagrams of string theory is the canonical example.
There are two angular coordinates on the torus. Imagine that we take both of them, $x,y$, to be in the interval $(0,1)$, with both end points identified with one another. Then a theory may have gauge transformations, namely diffeomorphisms of GR. The "small ones" are those like $$ (x',y') = (x+ \sin(2\pi x)/10, y+\cos(2\pi y)/5 ),$$ something that doesn't "essentially" change the way how the torus is parameterized. Those can be obtained from infinitesimal ones and the infinitesimal ones' behavior is encoded in the behavior of the currents – in this case, the world sheet stress-energy tensor.
However, the transformation $$(x',y') = (y,-x) $$ where I chose the minus sign just to preserve the orientation, for the case that the theory on the torus isn't left-right-symmetric, is qualitatively different. You can't produce it from the infinitesimal diffeomorphisms. This is just one example of a more general class of transformations $$ (x',y') = (ax+by,cx+dy)$$ where $ad-bc=1$ and $a,b,c,d$ are integers so that the periodic identification of $(x',y')$ with the unit periodicity is the same condition as the periodic identification of $(x,y)$. They form the group $SL(2,Z)$, the so-called modular group, and it's important because one has to check the invariance of a string theory under this modular group separately. This is actually a nontrivial condition that constrains the number of degrees of freedom, imposes level-matching conditions, forces string theories on orbifold to add projections simultaneously with the twisted sectors, determines the critical dimension of string theory, enforces the duty of lattices to be even self-dual in many cases, and so on. These constraints would be missed if we ignored the large gauge transformations. We would be thinking that some theories that are actually inconsistent are consistent.
On the other hand, the modular invariance for the torus already guarantees the invariance under large gauge transformations for higher genus surfaces. I wouldn't be able to reproduce the proof but I intuitively know why it is correct and I am confident that a rigorous proof exists.
Similar issues may appear for more ordinary gauge transformations, those of the Yang-Mills type.