One of the issues at the moment (besides the ethics and also the traumas of many failed attempts) is that they have not managed to clone primate babies from adults.
So if the process that was successful for monkeys was the only way it could work for humans, then it is not as though we could decide later in life to clone ourselves.
From the article: "The process is still very inefficient — it took 127 eggs to get the two babies — and so far it has succeeded only by starting with a monkey fetus. The scientists failed to produce healthy babies from an adult monkey, though they are still trying and are awaiting the outcome of some pregnancies. Dolly caused a sensation because she was the first mammal cloned from an adult."
One may feel uneasy, to say the least, reading that and some people may really freak out.
Another problem is that the new ability is viewed in the context of medical research - these genetically identical monkeys would be useful as research animals in labs.
It is not about preserving endangered species, for example.