Translation and context is everything. This site is excellent because it provides the Greek and it is possible to see how various translations of the text add color but can detract from meaning. The key words are βεβήλους (bebēlous) - translators use this for "wicked/heathen" but this can also lean toward "accessible/common." Then there is γραώδεις (graōdeis) - translators use this in a variety of ways, but we can see that this is the author being dismissive of things related to old women. Then there is an old, but also a false friend: μύθους (mythous). In the context of the time, we can regard this to mean a story or belief from the folk.

The author is warning people to avoid these elements of folklore - folk beliefs that are not attested to in the Bible. I'm not sure where you are heading in your question when you ask about "the same tales we hear today." - About Greek gods?; stories from Grimms' Fairy Tales?; stories told today about our modern world?

I'm not a Bible expert - and clearly there are people who devote their lives to understanding these sorts of passages - but from a folklorist's point of view, it seems to me that the author is warning people not to be taken in by the stories that circulate in the villages about the supernatural and extraordinary. He is suggesting that these are not the path to Truth and Salvation. These stories are not like the stories from Grimm, which were regarded as fictional and were told as entertainment. I also don't see this as the grand, huge stories about the "gods" - the stories that were the focus of civic obligation and community devotion.

Instead, it appears that the author is alluding to the sort of stories that we still hear today - stories about ghosts haunting a building or UFOs abducting someone, but it's hard for us to appreciate just how all-consuming these stories could be in the context of the day: they could reflect a persistent dread of the supernatural forces that lurked out there, every-ready to cause harm. It would be easy to become all-consumed by these stories until they dominated one's world view: perhaps the best modern analogy would be someone who listens to a steady diet of talk radio that feeds the listener with all sorts of conspiracy theories until that person's world view becomes clouded by that point of view. We get a glimpse of how these stories existed in their native context from Petronius's first-century Satyricon, in his description of Trimalchio's feast - particularly the stories about a werewolf and of witches abducting a baby.

I read 1 Timothy 4:7 as referring to these sorts of stories. Understanding these kinds of stories in a modern context is difficult but not impossible. I suspect today that Timothy would warn against being taken in by people who promoted the idea of conspiracy theories in that this can be a world-view that consumes the listener and shapes a world view that can be detached from reality. And more to the point - that can place a haze between a person involved with an element of folklore and the "Truth" offered by the Bible.