Turn Undead is really more channel positive or negative energy by the AD&D setting. At least, sort of. Undead existed in two planes simultaneously, the prime material and the negative energy planes. Turning via the focus of the cleric's holy symbol could affect undead in a blunt way. It could also affect extraplanar creatures too, so it wasn't entirely about being undead.
The difference between Arcane & Divine casting was that the magic user studied as an academician to use the power already inherent in the prime material plane like our historic alchemists would. They try and "unlock the secrets of the multiverse" and then speak, move, and use material to cast spells. Prepping was study and learning spells was done by studying the world, like how monsters work and the environment (including other inner planes).
Divine casters sought answers in their deity through meditation, prayer, fasting, or, basically, ritual. This is part of their training. Their preparations is reconnecting with those mannerisms. They prepared spells conveyed upon them from their deity. They channeled these and could choose as desired from what their deity offered. The spells were delivered via a focus, a symbol or whatever might represent the deity.
Turn Undead is the cleric (a term like magic-user or fighting-man that's meant to be very generic) channeling energy out in undifferentiated formlessness. It affects undead so well, because good cleric's positive energy harms negative, while negative energy bolsters itself. We reverse this when dealing with positive energy creatures like evil clerics turning devas or even paladins.
There are some unique features, like how monks are mystics who gain divine energy via meditation and bolster their own bodies and minds (the historic D&D brand being a type of shaolin monk, which could use some switching up a bit). Druids, who revere the natural world as deific, unlike all magic-users who study and dissect it, aren't channeling an other worldly power. They are like cleric's who have died and reside in their patron's demesne. They are part of the prime material world's power. They alter the world around them as almost as an extension of themselves. They aren't a gateway further and further opening to another realm, but an ongoing becoming that manifests their own.
All of this gets into the D&D cosmology, how magic works, and why there are certain division or differentiations peculiar to D&D. And the subclasses I listed are hardly the only ones possible. The key difference between arcane and divine though, is not (so much) a mechanical one. It is a matter of focus. Sword wielders are focusing on the fight, Wizardry is in mastering the multiverse, but clerics are in spreading the rituals of their deities. This could be planting crops for an agricultural god (lots of those in history) or spreading pestilence across the land.
There used to be an alignment/morale system for clerics to explore like fighters get the combat and wizards the magic systems. This included Followers too (henchman, hirelings, cohorts, and associates) But all of that is kind of unpopular nowadays, so I think the whole spectrum of the divine classes are being subsumed into the other two. Like others have said, a gish, which it could be if you drop all the cleric-y stuff.