There are a few caveats with the hodograph transformed solution. The first, already mentioned, is that the transformation is not one-to-one. In particular, the region of fluid that flows above the body, and the region of fluid that flows below the body usually transform to the same area in the hodograph plane. However, sometimes the transformation is more complex. There exist what are called ``limit lines,'' where the hodograph solution turns back on itself (i.e., the particle moves forward in the physical plane, but its path maps to a line that stops and reverses itself, in the process creating a multi-valued region). The effect is not unlike Riemann sheets.
Another problem is shock waves. Shock waves can present problems in the physical plane because they have very sharp gradients. Many times (especially with inviscid flow, I think), the gradients are considered infinite, and the shock undifferentiable. In the hodograph plane, it is much worse. A nondifferentiable shock in the physical plane is discontinuous in the hodograph plane. This would present much difficulty in computational solutions.
There are certain benefits of the hodograph transformation, however, which may make it worth the problems. First, as mentioned, it is suited to inverse design. Second, an infinite area in the physical plane maps into a finite area in the hodograph plane. This can reduce computational memory required by quite a bit.
But most importantly, when the flow is steady, isentropic, irrotational, inviscid, and two-dimensional, the transformed equation is linear. This is by far the most important reason for considering the hodograph transformation. The next section contains more information about this.