I think that beyond etymology even, Greek mythology still has a huge impact on our lives. Much of the linguistic and narrative vocabulary and tropes that we see in everyday media have origins in stories shared by Greek mythology and other historical sources of narratives, fables and lessons passed on over the years.
The names and attributes of Greek and Roman gods still hold sway over the modern world. On some level, we technology-obsessed beings still like or want to believe that superhumans or even supergods existed and played a role in shaping the past and hence the present state of this world.
Many of the Greek heroes' names are still in use today and many form the stem of words in current use. For example, we talk about somebody's weakness as being their "Achille's heel" because the Greek hero Achilles, was safe from harm so long as nobody attacked his heel. Hypnos, the god of sleep, is responsible for the term hypnosis. Zephyr, the gentle west wind gets its name from Zephyrus, god of the west wind.
Yes, Greek mythology still impacts our lives today, as do other ancient cultures. U.S. currency features a pyramid (Egyptian) and Latin (Roman culture) phrases. We use Greek phrases such as "herculean task" after the Greek hero Hercules. Movies and art depict Greek Mythology. Even business logos feature things like the Aegis Shield.