"Which book would you recommend to an experienced person of Greek Mythology for a better understanding?"I tried to answer it the way I suspected it was intended -- as well as the way it was written. Thus, the following recommendations are mostly for those inexperienced in Greek Mythology.
For somewhat older people, I would recommend Thomas Bulfinch's The Age of Fable or Stories of Gods and Heroes coupled with Ovid's Metamorphoses. Bulfinch is widely available, including online, and the stories entertain as well as explain, with the caveat that he prefers Roman names like Jupiter and Proserpine to Zeus and Persephone -- all explained in the introduction. Ovid's work is a classic that ties together so many stories as to be somewhat overwhelming, which is why I suggest combining it with Bulfinch, who, incidentally, developed many of his stories by translating Ovid. To be truly familiar with Greek mythology, you really should know a good portion of the allusions Ovid makes.
For those already familiar with Bulfinch, I would suggest Timothy Gantz' Early Greek Myths, although this is a 2-volume reference work, rather than a book to read. If you haven't already read The Iliad, The Odyssey, and Hesiod's Theogony, those are essentials for Greek mythology, and the works of the Greek tragedians, Aeschylus, Sophocles, and Euripides, are also basics. I find Euripides stylistically the easiest to digest.