If I do run a Forgotten Realms swords and wizardry Whitebox game, I want clerics of different gods to be mechanically different. Also, I plan to adjust the descriptions of different gods to match a L, N, C alignment system and my own vision for the realms.
Here are a few selected gods:
Auril - Chaotic - Goddess of Cold, known as the Frostmaiden
Weapons - as cleric
Armor - leather and hide armors of the north, shields
Clerics of Auril cannot use the Turn Undead spell,
+3 to saves vs cold attacks, +1 slot survival (tundra)
Auril is worshiped in the north, where the tribes offer up sacrifices to the frostmaiden to sooth her fickle rages that bring cold winds. Clerics of Auril are known to consort with frost giants, go to battle with spiked maces magically shaped from ice, and at higher levels can control the weather, call up ice elementals, maybe even ice demons...
Mystra - Neutral - Goddess of Magic
Weapons - as cleric, they favor the staff
Armor - none (as wizard)
Clerics of Mystra cannot use the Turn Undead spell.
Clerics of Mystra get +2 to saves vs magic, and can use magic items normally used only by wizards, they also get 1 extra slot in a lore of their choice (typically something related to magic), and +2 slots in read magic, they also have the ability to detect magic (as magical ability) once per day by concentration.
There are very few true clerics of mystra, most of her followers are magic users. Those clerics that follow her are usually found in libraries or as adventurers, seeking lost magical texts and artifacts.
Selune - Neutral - Goddess of the moon and stars
Weapons - as cleric
Armor - any, leather preferred
Clerics of Selune cannot turn undead, +1 slot Lore (navigation by the stars), they gain night vision (30ft). The priesthood of Selune is as scattered as the stars in the sky. They tend to meet in small groups out in the open or in buildings with open ceilings to allow view of the moon. Full moons are special holy days for the priesthood. Clerics act as seers, guides, mentors, using astrology to predict events. They are free spirited people fighting back both chaos and order when they oppose free will - trying to bring back the balance that allows. Folks outside of the faith are always distrustful of the order due to rumors of widespread lycanthropy in the priesthood (that are only partially true).
Lathander - Lawful - God of dawn, renewal, and birth, known as the Morninglord.
Weapons and Armor - as cleric
Clerics of Lathander can turn undead any number of times per day (once per round) as a special ability. Clerics of the Morninglord tend to dress in bright colors of orange, red, and yellow and many adventuring clerics favor tinted crimson or gold platemail. Their congregations meet for services in the morning and they tend to be short happy affairs with much singing and rejoicing. Clerics of the faith can call forth dawnglow (a rosy type of faerie fire), and have even been able to call forth focused bursts of pure light. Rumors have it that Lathander is only one facet of a more powerful sun god who may have other facets shown in some of the other faiths.
I have plans for even more gods, but the trend will be the same. Each faith will have customized spells and special abilities. Most clerics will either not be able to turn undead at all, or will have to memorize a spell to do so. A few selected dieties will offer the turning undead ability (like Lathander).