As you say, the left hand sides of the two systems are equivalent, because the matrix A in both cases has the same reduced form. So the two systems can only differ on the right hand side, which means that the b's can be different.
The first system might be
Ax = b_1,
and the second system might be
Ax = b_2.
[Edit: The following part has been pointed out to be wrong] Since the first system has an infinite number of solutions, the reduced form of A must contain a row consisting only of zeroes. For the system not to be inconsistent, the corresponding entry in b_1 must also be 0.
If however the corresponding entry in b_2 is non-zero, then the system Ax = b_2 will have no solution.
In conclusion, the two systems need not have the same number of solutions. The number of solutions depend on the vector b.