I have always been struck by what seems to be two descriptive gradations or levels of giftedness in French. I have seen "surdoué" used to describe children with abilities well above average. For example in mathematical ability this might be a fourth grader mastering calculus. But I have also seen sentences like, "Cet enfant est un pianiste doué." I've never been sure if the latter (doué)implies a less remarkable form of giftedness than surdoué. My sense is that a "pianiste doué" is simply an excellent musician, not necessarily uniquely gifted. But I have never understood the point at which "doué" becomes "surdoué." Are surdoué and doué nearly synonyms?
The two words are obviously related but nevertheless not synonyms.
Cet enfant est un pianiste doué, or better Cet enfant est doué pour le piano means the child has very good piano-playing abilities, that they quickly learned how to play the instrument and do it well. They are easily among the best players in their age group.
If we say Cet enfant est un pianiste surdoué, their abilities are exceptional. They surpass the performance not only of players of the same age but of much older pianists.
We also often just say Cet enfant est surdoué, or simply C'est un surdoué. We do not specify a domain here. We state that the child's IQ is well above the average, making them unusually skilled, at least potentially.
"Surdoué" is both more intense than "doué", and also typically reserved for young people who display abilities well beyond their years. An example might be Mozart composing original work as a child. While surdoué doesn't necessarily apply only to children, you probably won't hear it qualify people over the age of 30. You can also use surdoué as a substantive ("c'est un surdoué") which you cannot really do with doué.
On the other hand, someone can be "doué pour le piano", "doué pour la cuisine", etc. even in old age. It just means they have better results than most people given the amount of work that they put in - they make it look easy.
Surdoué doesn't only imply very gifted, but somehow too much gifted. If you said that about a child's level in mathematics it implies a problem with the situation: that child should be moved to the next grade, they get bored in class and get disruptive, etc.
That said, many people just use it as an hyperbole for "very gifted".