A farmer who has never had the means nor the opportunity to travel much farther than the next village would likely describe the incantations of a local witch as "the devil's tongue". A pale scholar who sits in a library tower and philosophises pretentiously over various texts and theorems and calls himself a wizard would describe these utterances as "arcane" or "spells". To each, this is a strange speech, its true meaning lost to time.
Both are wrong. There is little difference between the language used to bring about magics and the common languages in which mortals speak.
A wizard will learn many spells through their studies, most often starting with a simple command to create a ball of flame. What wizards insist on calling a spell, however, is little more than a sentence - badly pronounced due to centuries of being written and rewritten without any scholar ever bothering to retrace the grammar back to its source. A spellbook is a phrasebook, and wizards are tourists, reciting commands phonetically.
In the case of Fireball, the sentence is spoken in the language of fire, and its words are an instruction to the spirits of flame within the air to burn and then falter. These are simple words, comprising most of fire's language, and so the command is easily understood, despite the rough accent of humanity. Fire longs to burn, and falters when it must, and so all the command requires is a tone of authority for the spirits to heed it and obey. One thing wizards have never lacked is a sense of self-importance, and so that commanding tone comes naturally - this "spell" is considered an easy one.
The magic used by thieves is of the same make, though wizards would never acknowledge or understand this. When a hand-mage picks a lock with a spell, she is simply reciting a sentence in the language of trickster spirits, passed down from thief to wily thief, promising a honeyed seedcake if the spirit will indulge the speaker in spreading a little chaos. That thieves make regular offerings to trickster spirits at shrines is seen as a separate tradition, its purpose misremembered.
Some spells are more difficult, this is true, but not for the reasons scholars believe. The spell to command storms, for instance, is considered far more advanced and powerful than conjuring a simple fireball. There is however no inherent difference in the incantations themselves, beyond the fact that a cloud's language has more complex grammar than a flame's. Where the difference lies is in an area wizards pay little heed to: listening, as well as talking.
It is not enough to learn a sentence and speak it to command a storm. A conversation must be held, with the speaker paying close attention, involving themselves in the politics and passions of the sky. Contrary to popular understanding, storms care little for where their rain may fall or their lightning may strike - what's important to them is what's happening up there. To truly conjure a storm, the wizard must speak with the cloud on its level, heart-to-heart, and empathy is not a thing that is taught in wizarding schools.
(It is true of course that spells of "command storm" exist, recorded on scrolls in flowing glyphs, lauded by scholars as things of great power for only the most adept mages. In truth they are nothing more than long-winded diatribes that pay no heed to their audience's temperament. A storm does not come because a wizard's spell commands it, but because after a minute or so of chanting and hand-waving the storm gets bored of the little man in the funny hat who suddenly started speaking its language, and wants to get the whole thing over with. Wizards being struck by stray lightning bolts is the storm's aim, not a side effect.)
The magic of hedge witches is seen as backwards, rural, too overly concerned with small things and not nearly high-minded enough for scholarly pursuit. The truth is that a witch is simply someone who takes the time to learn the languages of the local spirits, beyond simply reciting phrases and barking orders. She devotes time and attention, forms relationships, makes requests and gives back in return. She does not simply recite a command from rote memorisation when she makes a fire appear, but instead greets the fire spirit with a song, asks it how its day is going, welcomes it into her hearth.
Common folk and scholars alike are suspicious of how witches will live in one place all their lives, away from society. They simply can't get their heads around the idea that the woman living alone in the hut by the forest is not lonely at all, but surrounded by friends with whom she communes and converses each day. A witch's house will famously attack intruders of its own accord, and this is no myth - the spirits that dwell there with her are simply looking out for their friend.