It's a bit hard to read the diagram, but I assume the length of ZW is given to be 1.

Statement 1 is not sufficient, because we have no information about angle b, so about where point Z is, and that information is crucial. If Z and W are very close together, a line of length 1 is very short in the diagram, so XY would be very long. If Z and Y are close together, a line of length 1 is very long in the diagram, and XY will be quite short.

Statement 2 is sufficient. If a - b = 0, then a=b. So triangle XYZ is isosceles and XY = XZ. Further, around point Z, we have a straight line, so the angles add to 180, and angle XZW is 180 - a. So in triangle ZXW, we know two of the angles are 180-a, and a/2. Since the three angles in that triangle must sum to 180, the missing angle, ZXW, must be a/2. But if the angles in ZXW are 180-a, a/2 and a/2, then ZXW is also isosceles. So ZW = ZX = XY, and if ZW is 1, so is XY.