In classic folklore studies, those similarities were seen as evidence of the antiquity of many of the folktale "types." In recent decades, the validity of "types" has been disputed by some, but the concept remains as the means to organize folklore archives and is still used by many folklorists as a means for comparative analysis. That tedious caveat is necessary because any answer given to your great question triggers the tripwire that some students of folklore feel exists when one mentions "types."

So that said ... European folktales have long been recognized to fall into "types" - plots that are repeated across expanses of geography and in some cases with considerable depth historically. These were initially organized in the early twentieth century in a "type index" by the Finnish folklorist, Antti Aarne, and his index was later edited and enlarged by the American folklorist, Stith Thompson: The Types of the Folktale: A Classification and Bibliography (Helsinki: FF Communications No. 184, 1961; second revision, fourth printing, 1987). This was more recently edited and enlarged by the German folklorist, Hans-Jörg Uther: The Types of International Folktales (Parts I–III) (Helsinki: FF Communications No. 284, 285, and 286, 2011).

With this tool, we can consider three examples of pre-modern folktales (appearing in print as "fairy tales") that enjoyed wide geographic distribution and also appear in ancient sources. For example, ATU (Aarne-Thompson-Uther) Tale Type 313, The Magic Flight, has episodes found in common with the ancient Greek text of Jason and the Argonauts. Similarly, ATU Tale Type 300, The Dragon-Slayer finds ancient Greek expression in the story of Perseus and Andromeda. Many see this same story expressed in the rescue of Princess Leia from Darth Vader (the dragon can assume many shapes!). Some stories seem immortal!

An astounding expression of a folktale appearing in ancient text occurs in an Egyptian papyrus dating to roughly 1250 BCE that features the story of "The Two Brothers" - ATU 303, "The Twins or Blood-Brothers." This offers primary source documentation of a folktale "type" with a history spanning over three millennium.

If we can accept these similarities as meaningful (and I do despite the criticism of some), then the question remains, "why are there ... so many similarities between older myths and more recent fairy tales?" - as you ask.

The answer to the "why" of it (my examples only address the "what" of it), revolves around the strength of persistent traditions. The peculiar things about folklore is that it is both tenacious and flexible. It can be enduring and yet there is a fluidity in oral tradition that makes change inevitable. Folklore in its natural oral setting perpetuates stories and traditions, passing them on from one generation to the next. At the same time, the effect of the individual on tradition as it passes from that person's lips to another's ears can cause change. How this "push and pull" in oral transmission over generations occurs has been the subject of study for two centuries, and there are no really good explanations for this other than to describe that process and understand that it is indeed a process.

Another aspect of your question centers on the issue of belief. Folktales (or fairy tales) were traditional, told in pre-modern Europe as the popular novels of the peasantry: they were generally told as adult fiction. That's distinct from legends, which were generally told to be believed. And yet when we reach back in time to ancient myth, we encounter stories that were apparently told to be believed: people were meant to believe that Jason and Medea really did flee from her father and that Perseus really did kill the sea monster to rescue Andromeda. The situation with the Egyptian "Two Brothers" is less clear in this regard. All we can deduce from this conundrum of genres is that some stories seem able to slip effortless across the folktale/legend divide from one place to the next or one time to the next. This is no big deal, but it is worth noting.

Not sure if this answers your question - feel free to let me know how unsatisfactory my answer has been and I'll try to do better!