The domain represents the possible values that x can be. In mathematics, you cannot divide by zero. If you plug in 0 to f(x), you will be dividing by zero.
f(0) = 0/0
f(x) = g(x) for all values of x except 0.
Therefore, the difference between the domain of x/x and 1 is that 1 is all real numbers, and x/x is all nonzero real numbers.