X-rays are a useful and fairly inexpensive diagnostic tool to screen for a wide variety of problems that can cause pain, swelling, and loss of motion. As you probably saw from your own X-rays, calcium deposits often (but not always) show up clearly. Thickening of the surrounding soft tissues can also be seen on X-ray.
More advanced imaging studies such as CT scanning and MRIs can offer additional information to guide treatment. In the case of calcific tendinitis of the neck muscles, the additional tests will help rule out infection as a cause of any swelling. Infection and inflammation are two separate problems that are treated with different medications.
An abscess from infection in the neck can be a potentially dangerous problem. CT scans and MRI can help pinpoint the exact location of the problem. The results also show the doctor what's going on in the soft tissue structures.
Doctors are usually very conservative when it comes to ordering extra tests. Given the added cost of these more advanced imaging studies, they are only suggested or ordered when the information gained is important to your care and treatment.