In chapter 3 of Hartshorne's Deformation Theory, he defines functors $T^i$ for $i=0,1,2$ that take as input a ring homomorphism $A\rightarrow B$ and a $B$-module $M$ and outputs $T^i(B/A,M)$, a $B$-module.
Among the properties given are that a short exact sequence of $B$-modules gives a long exact sequence of the $T^{i}$'s and $T^0(B/A,M)={\rm Hom}_B(\Omega_{B/A},M)$.
What is the difference between the $T_i$'s and taking the derived functor associated to ${\rm Hom}_B(\Omega_{B/A},\cdot)$ and does this difference measure anything?
The construction first creates a complex $L_2\rightarrow L_1\rightarrow L_0$, where $L_1,L_0$ are free and give a resolution $L_1\rightarrow L_0\rightarrow \Omega_{B/A}$, so I think $T^0$ and $T^1$ agree with the derived functor cohomology, but I don't know about $T^2$.