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.
I admit I am a bit confused by your terminology, but here is how I learned it: Let $P$ be a $G$-principal bundle and $\Sigma$ a spacetime.
- gauge group: The fibers of the $G$-principal bundle over the spacetime, i.e. the group $G$.
- (Local) group of gauge transformations: The group of diffeomorphisms $t : P \rightarrow P$, which are fiber-preserving and $G$-equivariant, i.e. if $\pi : P \rightarrow \Sigma$ is the projection then $\pi \circ t = \pi$, and $t$ commute with the group action on $P$.
One can now, by transitivity of the group action on the fibers, define a function $g_t: P \rightarrow G$ by $t(p) = pg_t(p) \forall p \in P$, and such functions $g : P \rightarrow G$ conversely define a gauge transformation by $t_g(p) = pg(p)$ as long as they fulfill $g_t(ph) = h^{-1}g_t(p)h \forall h \in G$, so we have two alternative characterizations of local gauge transformations:
$\mathcal{G} = \{t |t \in \mathrm{Diff}(P) \wedge \pi \circ t = t \wedge t(ph) = t(p)h \forall h \in G\} = \{ g| g \in \mathrm{Maps}(P,G) \wedge g(ph) = h^{-1}g(p)h \forall h \in G \}$
The equivariant diffeomorphisms of $P$ are called local, since they apply a different group element to every spacetime point.
Now, the associated bundles are affected as follows: Let $\phi : \Sigma \rightarrow P \times_G V$ be a section of the associated bundle, i.e. a field. By a similar argument to the above, these are in bijection to $G$-equviariant functions $f_\phi : P \rightarrow V$ satisfying $f_\phi(pg) = \rho(g^{-1})f_\phi(p)$. This is esentially the reason why, in $\mathrm{U}(1)$ symmetry, a gauge transformation $\mathrm{e}^{\mathrm{i}\alpha(x)}$ acts on fields as $\phi(x) \mapsto \mathrm{e}^{-\mathrm{i}\alpha(x)} \phi(x)$.
So, you see, the local group of gauge transformations is much bigger that the global gauge group since it allows far more functions than just the constant ones. You can always clearly write down the global gauge group (it defines your theory!), but writing down the local one more explicit than I did above is hard. For $\mathrm{U}(1)$, however, it is just $\{x \mapsto \mathrm{e}^{\mathrm{i}\alpha(x)} x | \alpha : P \rightarrow \mathrm{U}(1) \text{is smooth (enough)}\}$, I think. Cases where the two groups coincide demand a spacetime that is a point, I would guess, but I am not wholly confident in that.
Also, all of this can be done classically, nothing about gauge theories is inherently quantum.
EDIT:
Alright, your edit was very helpful in discerning what is actually going on here.
Your global gauge group is what physicists call the group of gauge transformations. The gauge group of a a gauge theory is what you call a local gauge group (and what the nLab also calls the local gauge group). When physicists say the gauge group $\mathrm{SU}(N)$, they mean it is what you call the local gauge group.
The global gauge group of the nLab is just the group of transformations (not necessarily gauge transformations, terminology is terrible here, I know) that leaves all observables invariant, i.e. it is the group of symmetries of the theory (not the group of symmetries of the Lagrangian), the group of gauge transformations is naturally a subgroup of this. The difference is that this global gauge group can contain transformations that have not really something to do with the structure of the local gauge group, and can contain things which are not gauge transformations. This global gauge group can even exist if you have no explicit gauge theory, and is inherently a QFT concept.
In other news, you are right, your connection form $\omega$ is the gauge field $A$ of a physical gauge theory, and it transforms exactly like you wrote. Now, the problem with the gauge field is exactly that ugly transformation, so we construct the curvature transforming in the adjoint rep and call it the field strength $F$. The action of a pure (Yang-Mills) gauge theory is then (up to prefactors) given by
$$ \int_\Sigma \mathrm{Tr}_{ad}(F \wedge \star F)$$
since the action must be invariant under gauge transformations and the $\mathrm{Tr}_{ad}(F \wedge \star F)$ is pretty much the only object we can construct out of the gauge fields that is invariant and can be integrated over the spacetime.
Best Answer
In the cases when the gauge group is disconnected, both choices of defining the physical space as a the quotient of the field space by the whole gauge group $\mathcal{A}_{physical} = \frac{\mathcal{A} }{ \mathcal{G}}$ or by its connected to the identity component $\mathcal{A}_{physical} = \frac{\mathcal{A} }{ \mathcal{G}_0}$ are mathematically sound. In the second case, the large gauge transformations are not included in the reduction, thus they transform between physically distinct configurations., and in quantum theory between physically distinct states.
However, as N.P. Landsman reasons, the first choice overlooks inequivalent quantizations that correspond to the same classical theory. In the case of the magnetic monopoles these distinct quantizations correspond to monopoles with fractional electric charge (Dyons). This phenomenon was discovered by Witten (the Witten effect). If the whole gauge group including the large gauge transformations is quotiened by, no such states would be present in the quantum theory.
In the monopole theory, the inequivalent quantizations can be obtained by adding a theta term to the Lagrangian (just as the case of instantons). Landsman offers an explanation of this term in the quantum Hamiltonian picture: Assuming $\pi_0(\mathcal{G})$ is Abelian, then when the gauge group is not connected, then a gauge invariant inner product can be defined as:
$\langle \psi| \phi \rangle_{phys} = \sum_{n \in \pi_0(\mathcal{G})} \int_{g\in \mathcal{G_0}} e^{i \pi \theta n} \langle \psi| U(g) |\phi \rangle$
Where the original states belong to the (big) gauge noninvariant Hilbert space. This inner product is $\mathcal{G}_0$ invariant for all values of $\theta$.