The Difference Between Arcane and Divine Magic

Divine Magic, aka Miracles Deities are individuals from a parallel plane of existence called the Celestial Plane, who gain more power the more belief they garner. They are made out of a great deal of psychic force, collected from the minds of the creatures on the material plane. It is for this reason they interfere with the events therein.
They have found that they can best gain new followers (and therefore power) by having agents on the material plane willing to enact their miracles, in a way that is in character for the deity. Basically, a god gets very busy doing their god things, and doesn't have time to weigh the pros and cons of every possible holy act. For this reason, they choose champions (like clerics, paladins, and priests) to make those nitty gritty choices for them. Then, when the time comes, the mortal can channel the power of the deity into an act of miracle - this is colloquially called a spell, which is misleading. It's more like asking a favor from someone who's much to busy to care what the favor actually is, and just grants it with a swish of their hand. 
So; the gods trade occasional miracles for belief and the promise of more believers, which serve as a kind of faith-based sustenance.
True magic, arcane magic, is very different.
Arcane Magic Across the entirety of reality, there is a fine web of chance and reality. By pulling at the threads of this web in practiced, precise ways, unlikely things occur. These are spells. What you have just read are the utmost basics of spellcasting. It is, of course, much more complex.
What is the web made of?  That is an excellent question, O theoretical reader! There are many schools of thought that argue on this point, but a general consensus is: we have no clue. It is invisible and intangible to most, stretches across every plane, and holds the past, present, and future in it's strings.
Why can't I see it? Because you have not been trained to, is the short answer. The first thing a new spellcaster learns is how to perceive the web (a little cantrip called Detect Magic), which involves intense focus and can result in a lot of burnt retinas. If you'd like to try your hand at it, keep your eyes open for at least 2 days straight, with the assistance of eye drops concocted from lich sweat and unicorn urine, and try your best to look closely. After this gruelling procedure is over with, you will be able to activate that state of the eyes on a whim, but more focus will be necessary for a more precise view.
Once you can see it, you can learn how to reach out and touch it. Most wizards can feel the web constantly, even if they aren't visible, and to be able to cast by feel is a very important skill. Then, you can reach out to a passing strand and give it just the right tug or twist, and BAM - there's a fireball.
How does pulling a weird invisible string make magic happen? Now this is a controversial question indeed. There is a near even split in the realm of arcane academia about the precise functions of the web, which are Probability Theory, and Planar Theory.
Probability Theory is based on the idea that the strings are made of some kind of concentrated chance, or luck. By wiggling them around, you can increase the likelihood of, say, a spontaneous fireball, or an enemy deciding Hey, these guys aren't so bad after all. Opponents of this theory claim there are some things that are just impossible that can be achieved through magic, and that no string of chance or fate could cause something like that to occur. Another argument against this idea is that some wizards, despite their skill, are quite unlucky.
Planar Theory is also called spatial theory or elemental theory, because it believes that by plucking at reality's strings, you are actually connecting two disparate locations. For a fireball, you are connecting your location to the Plane of Fire. For levitation, the Plane of Air. For a scry on another specific locale, obviously that place's neural impression is being blipped into the caster's brain. This theory holds up in almost all ways except charms; mind control magic just doesn't make sense within this concept.