This is a huge topic. There's no way to talk about it adequately in 2000 characters.
I don't think much of Greco-Roman mythology has a <em>direct </em>influence on modern Western values. However, many commonly understood terms and sayings ultimately have their roots in the old stories.
You may warn a proud and overambitious friend not to <em>fly too close to the sun</em>; this saying derives from the story of Icarus, whose wings, glued together with wax, melted and fell apart as a result of flying too close to the sun.
Narcissism is a negative personality trait that derives its name from Narcissus, who fell in love with his own reflection. The best-known version of the story has him stare so long into a pond that he became a flower (now called a narcissus flower), but in another version he commits suicide because he cannot have the object of his desire.
The philosopher Plato had one of the biggest influences on Western philosophy until modern times; it has been observed that "all [Western] philosophy is just a series of footnotes to Plato." He also used and invented myths to illustrate his ideas. Many of these are fairly obscure, but at least one is recognizable to almost everyone in the West: the myth of Atlantis.
The modern French philosopher used the story of Sisyphus as a potent symbol his concept of the Absurd.
Note that not all of Western mythology comes from the Greco-Roman tradition. The Norse, Celtic, and Slavic mythologies are distinct and influential. Depending on exactly what you mean by 'mythology', you could also count the Judaeo-Christian tradition, which I would argue had a bigger impact that Greco-Roman mythology on modern Western values (including secular values).