The major advantage of open market operations is that they inject money directly into the economy (or they extract money directly from it).
When the Fed conducts open market operations, it wants to be able to have an impact on the money supply. When it does open market operations, it gets to affect the money supply directly. It does not have to rely on people to borrow money (as with interest rates). The Fed simply injects money into the economy by buying securities or extracts it by selling them. This means open market operations can have a direct impact. One other advantage is that these operations can be done on a daily basis and can therefore be used to make fine calibrations to the money supply where interest rate changes cannot be.