Actually, "The Art of War" is only the title of the first western translation of Sun Tzu's work, from 1772. It was done by a French missionary, Joseph Marie Amiot, who was at the time official translator of Western languages for the Qianlong Emperor and lived in Beijing. The original title's meaning (兵法) could be given in English more directly as "military ways". But the first translation was French, and the translator choose to name the book "Art de la Guerre", ie "The Art of War". Due to the subtle nuances of each language, translators most often would try to convey the inherent meaning of a text, rather than match word for word. And Machiavelli's book of the same name was already well known at the time. He had been translated into French, German and English and was frequently quoted as authority. In 1757 came out another similarly titled book, also influenced by Machiavelli - "Reveries upon the Art of War", authored by Marshal de Saxe, a famed German military leader.

Machiavelli lived two centuries before Sun Tzu was translated into European languages, which makes it highly unlikely that he was influenced by the other work. His "Art of War" ("Dell'arte della guerra") was written 1520 and so far there's no evidence that he, or anybody else in Europe, had access to the original Sun Tzu text at that time.

The context of the two books is also different. While Machiavelli is mainly concerned with the philosophy of war, Sun Tzu provides very practical strategies and tactical guidance which can be used in any sort of competitive confrontation. There are similarities in the way of thinking but that's more likely to be a result of the similar logical paths the two authors followed in order to reach their conclusions. For example both Machiavelli and Sun Tzu expressed a belief that war and politics went hand in hand. For Machiavelly war was a political instrument and politics was warlike; for Sun Tzu war was of vital importance to the State, leading to either safety or ruin. But their opinions often differ: to Sun Tzu war was politics by violent means and the destruction of a society is not an objective in itself; for Machiavelli various causes of war led to various kinds of wars, ie the objective of the Machiavellian wars can be unlimited, including the case where entire people would aim to overthrow the society of another people.

R.Cantrell's "Understanding Sun Tzu on the Art of War" is a short introduction to that work, while E.Benner's "Be Like the Fox: Machiavelli in His World" explains how the times influenced the man's life and thinking.

According to /u/Platypuskeeper, it was a coincidence. Machiavelli would not have had access to Sun Tzu's work.