Projective Representations
When considering the Lorentz reprensentations $(\frac{1}{2},0)\oplus (0,\frac{1}{2})$ The rotation around the $z$-axis is given by $$\Lambda_s(\Theta_z)=\exp(i\Theta_zS_{12})=\begin{pmatrix}\exp(\frac{i}{2}\Theta_z) &&&\&\exp(-\frac{i}{2}\Theta_z)&&\&&\exp(\frac{i}{2}\Theta_z)&\&&&\exp(-\frac{i}{2}\Theta_z)\end{pmatrix}$$ leading to $\Lambda_s(2\pi)=-1$.
This should not happen in the Lorentz group! Rotations around $2\pi$ should always map to the identity (See e.g. $4\times 4$ matrix representation).
The reason that this happens for spinors is that by exponentiating the Lorentz algebra we do not only get the Lorentz group but the universal cover of the Lorentz group $SL(2,C)$.
So spinors transform under $SL(2,C)$ and not the Lorentz group.
In field theory, observables should not depend on a complex phase. Thus, for physics we are not only looking for representations fo the Lorentz group but so called projective representations. Although the Spinors transform under this more general group, the physical observables will still respect poincare invariance!
Definition: Projective Representations
A projective representation for a group $G$ is a representation on some representation space $V$ $f: V\to V$, which fulfills $$f(g_1)f(g_2)=e^{i\phi(g_1,g_2)}f(g_1g_2).$$
This is slightly more general than the normal requirement for a representation.
The projective representations of $O(1,3)$ are the same representations as the normal representations of $SL(2,C)$.
The reason that something like the spinor representation exist is that e.g. the group $SO(3)$ is not simply connected. That means there are closed paths throughout the group manifold which can not be smoothly deformed into a single point. This is usually due to some type of holes inside the group. The universal cover of the group in contrast will be simply connected.