On the surface Witchcraft Earth is just like our Earth. The overall political, technological, sociological, and cultural situations are identical to our world. Language, music, literature, world leaders, to pop culture are all the same as ours. There is only one difference that separates our two worlds...and it is a significant difference. On Witchcraft Earth, magic and witchcraft are not simply fantasy...but real commonplace aspect of everyday life all around the world. Magic and mundane co-exist. Those with magical ability have always had a significant place within most societies. The data for this can be found in many ancient texts; including the Bible, Qur'an, Talmud, Dead Sea Scrolls, Book of the Dead, etc.
Civilization on this alternate Earth is a fusion of 21st century life and the wonders of sorcery. Especially in the United States. On Witchcraft Earth, the Salem Witch Trials never happened. As a result, the slow to grow witchcraft movements made their way through the elite circles in Colonial American society. During his first term, President Thomas Jefferson even publically confessed to being an accomplished wizard.
The phenomenon called "magic" involves mentally accessing and controlling certain energies and entities from dimensions tangential to our own. Because these other dimensions possess physical laws that are tremendously different from our own, the manipulation of the extradimensional forces gives people the ability to perform a variety of acts and displays that defy the universal laws of physics as well as manipulate and alter many aspects of reality on a fundamental level. Simply put, magic is the process of using powers from other universes to reshape ours on intrinsic levels. Although compared to the scope of the entire universe, it is within a relatively microscopic and localized area.
The trait of the magical ability is purely a genetic difference and there are a variety of genetic factors controlling the it. There is no single gene for the ability, instead it is scattered throughout the genetic material and the combination determines magic using strength. There is a spectrum of magical abilities in humanity, ranging from no discernible magical ability even with training, through magical abilities only manifested with training, up to spontaneous magical manifestations. The majority of humanity falls into the middle category.
Almost all magic is done with the use of a supporting tool, typically a wooden wand. Wood is a magically imbued material. This is because in addition to harnessing sunlight, the photosynthetic process also has a resonating quantum effect that slightly augments the special protein structures that allow people to perform magic. Wands function as a "magic focuser". By using a wand one's magical powers are greatly focused. In the same way a convex glass can focus sunlight enough to create a small fire, a wand focuses the magical potential of its user, not necessarily increasing the magic power, just making it more intense.
Furthermore, most actual spell casting is done by using short incantations (most often in what sounds like a modified form of Latin) accompanied by gestures. This is because certain words and gestures act as psychological triggers to alter the perceptions of the mind, allowing one to summon an extradimensional force to override the usual laws of nature. Potions and talismans also serve as psychological conduits to summon a power. Accomplished magic users can sometimes perform magic, especially simple magic, without the need for any sort of conduit.
The existence of the Neanderthals is the evolutionary pressure that caused the magic-using genetic traits to evolve in Homo sapiens sapiens. Humans won out and the Neanderthals died out when modern man appeared on the scene because many humans had magic, while not a single Neanderthal possessed that gift.
With no spells or wand of usage the magic of these primitive magic wielders manifested their abilities rather subtly, it appeared as frequent cases of good luck or happy accidents. However, some of these people acquired some rudimentary control over their power--enough to aid in the hunt and in causing or curing various simple maladies. With this advantage; Homo sapiens was able to supersede the Neanderthal and flourish; driving the less fortunate primitives out.