One of the big differences between the Realms and Core D&D is the prevalence of deities. There are no generic clerics/paladins/druids or ones who worship an idea/concept. If you cast divine spells, you have to have a patron deity, it's just a hard requirement of the setting.
Almost everybody has a patron deity, even if they aren't devout, they have at least a little faith in a deity. If you die in the Realms without a patron deity, or having completely fraudulent worship of your patron, you only have a few days to get Raised or Ressurected, after that it takes a Wish/Miracle to bring you back (as the God of the Dead, Kelemvor, disposes of your soul since no other god will take it). When you die, your spirit goes to his realm for processing, where it is kept around for a several days as servants of deities arrive to take you on to your final reward (or punishment), and if nobody ever shows up, your spirit is disposed of by being turned into a brick and used as building materials for his realm, and at that point a Wish/Miracle is needed to get you back (to cajole a deity to send a represenative to fetch him).
This isn't as limiting as it sounds, because there as a lot of deities in the Realms (especially when you count in all the various demigods), of all alignments, with a wide variety of domains (including a list of new domains specific to the Realms), and some deities will take almost anybody as a worshipper, regardless of alignment.
Also, arcane magic is common. Not everyday-common like Eberron, but more common than Greyhawk. There are a lot of powerful wizards in the world, most kingdoms have at least one epic wizard (or sorcerer), sometimes more. The typical commoner is still pretty distant from magic in most places, but to adventuring folk you'll run into a lot of magic.
There are other little things, like Mage Sigils (each arcane spellcaster has a unique glyph that acts as a signature), the Goddess of Magic, Mystra, ensures that no two spellcasters use the exact same Sigil (through divine inspiration) and if anybody ever tries to forge anothers sign for fraudulent purposes (like forging a signature) they are likely to be struck down with a divine curse.
Magic is a fundamental weave that permeates the world, maintained at all times by the goddess of magic, and she balances out the huge demands and drains on it placed by users of magic. There are places where this weave is damaged (as a result of huge magical cataclysms), and these are Wild Magic zones (where magic can randomly misfire) or Dead Magic zones (permanent antimagic shells). She can deny any spellcaster the use of magic, but rarely uses it, and while theoretically possible for her to deny all divine casters their power, by divine agreement she does not interfere.
There is the Shadow Weave, another weave of magic created by Shar, goddess of shadows and darkness (and Mystras enemy), that gives special power to Illusion and Necromancy magic, and penalties to Evocation. Those who take up the Shadow Weave (which requires a feat) go a little insane (permanent Wisdom loss unless Shar is your patron deity), and forever cannot touch the normal weave. It is corrupting and twisting, and beyond Mystra's control (Shadow Weave magic cannot be normally dispelled or detected for example). It's a relatively rare thing, but very dangerous. A caster who does this is also placing themself under the jurisdiction of a dark and capricious deity.
But yeah, find an experienced DM, talk to him, learn by playing and you'll go a long way. No two DM's run the Realms the exact same way, but these are some of the differences in the Realms from Core D&D.