Similarity 1: Intelligence still modifies the number of spells you can learn (3e gives bonus spells learned for free, DCC's increases the maximum number of spells you can have at once in your grimoire, which doubles as the number of spells you presently are able to cast), and the highest level spells you can know/cast.
Similarity 2: Spells can be lost after being cast in both systems. In DCC it's not guaranteed, while in 3rd ed/PF it is.
Similarity 3: Many spells are superficially similar, like flaming hands, or sleep.
Difference 1: In DCC you will often find yourself rolling an Int check to see if you can learn a given spell you've been exposed to this level. Sometimes you won't be able to learn it and will have to wait a level until you can try to learn it again.
Difference 2: In DCC by default you never get to "pick" your spells to learn - you can only learn them through exposure in the game fiction, and certainly not by paying somebody in coins (though rare treasures like wizard's staves, wands, weapons with plusses, and the like may suffice; a demon's quid pro quo may be something that makes you question if a spell is worth having at all!) like the 3.5 PHB suggests may be possible.
Difference 3: Hot Hand Vancian is probably the best descriptor of DCC's magic system. You can lose spells for the day, which must be re-memorized, but only if you fail a spell check (d20+int mod+CL). The greater your spell check the greater your spell's result, which you find by referencing a given spell's list of potential results based on your check result. As long as you keep making your spell checks you can keep casting a spell. Spell checks can crit and fumble just like weapon rolls, though what happens on a crit and fumble is slightly different.
Difference 4: Peculiar effects abound. Each spell a wizard learns has a Mercurial Magic effect that occurs each time he casts it - these can be positive to quite bad, but they are relatively unique. When combined with a randomly determined or selected appearance for a spell when cast (each spell gives usually at least 3 or 4 options for spell appearance) and the varying levels of result from spell check, even the same spell is unlikely to look identical when cast by two different wizards.
Difference 5: Critical failures on spell checks cause misfires or corruption of the wizard's physical or mental form. A wizard with a crab claw for an arm is probably going to show up if you play for long enough, among other things.
Difference 6: Spellburn. You can temporarily burn points of an ability score to gain a bonus to your spell-checks. For particularly powerful or evil magic this may be required by the spell itself in some amount; for instance making a staff with the Wizard's Staff spell requires 10 points of spellburn minimum.
Divine magic is like arcane magic, just with the following differences -
Spells can be continuously cast until you roll a natural failure, which begins at nat 1 and whose range increases each time your spells fail to take effect (like the wizard you need a reasonably high spell check for your spells to work). Rolling in this natural failure range results in your deity being displeased with you (in a reverse of cause and effect - your deity was displeased with you all along, which explains all the failed spells, but you only realize it when rolling inside this natural crit failure range), and a subsequent roll on a Deity Disapproval chart to see what strange task you're given to prove your continued value as a cleric worthy to receive divine aid. An hour of prayer is on the low-end, while the high end includes tasks like an inability to turn unholy creatures for a period of some days, or other such issues. This disapproval range usually resets overnight (as opposed to resetting your spells for the new day like in 3e D&D), but sometimes it doesn't for particularly egregious transgressions. Rolling a natural failure doesn't cause you to lose your spells (necessarily), so unless the deity disapproval result says otherwise you can just keep casting any spell you know whenever you like - and probably dig yourself into an ever-deepening hole of deific obligations, but hey, life as a cleric.
Clerics also have access by default to Turn Unholy (their alignment determine what unholy creatures may be turned - for instance in addition to undead a lawful cleric may turn demons, devils, mythical monsters, chaotic dragons, etc.) and Lay on Hands, both of which function in the exact same method as their other spells - failure increases disapproval range, but you can continue casting them until the cows come home.
Clerics really can't use spellburn, generally speaking.
That should just about cover it, I think.