Difference between these two integrators
There are different methods to approximate integration in discrete time. The most straightforward ones are the forward and backward Euler methods, and the trapezoidal method. A discrete-time system with transfer function

$$H(z)=\frac{T}{z-1}\tag{1}$$

implements the forward Euler method.

Similarly, the backward Euler method is implemented by

$$H(z)=\frac{Tz}{z-1}\tag{1}$$

Finally, the trapezoidal method is implemented by

$$H(z)=\frac{T}{2}\frac{z+1}{z-1}\tag{1}$$

These discrete-time integrators are also described on this mathworks page.