From Diablo Wiki
"Spells" is an informal term for skills that are cast as magical attacks, rather than skills that modify weapon-based attacks. Spells are largely used by mage characters; the Wizard and Witch Doctor in Diablo III.
The term is nebulous; virtually everyone would agree that skills like Electrocute and Locust Swarm are "spells" but skills like Ground Stomp and Impenetrable Defense might not be considered as such. There's no game importance to the term; it's purely a piece of terminology.
There are several major changes to the calculation of spell damage in Diablo III.
- There are no skill points.
- There are numerous changes to attributes.
- Spell damage is determined by weapon damage.
- Spell casting rate is derived from weapon attack speed.
- Items add +% spell damage. (Removed during development.)
- Runestones vastly increase skill variety.
These factors combine to greatly modify the performance of spell casting characters in Diablo III. See the following sections for more details.
The removal of skill points is one of the biggest of the many changes to the combat engine in Diablo III. Characters get one skill to start with, and additional Active Skill Slots become available at Clvl 3, 6, 12, 18, and finally 24. Characters thus have a maximum of six skills enabled past Clvl 24, though the six so chosen can be switched around endlessly ((see the respecs article for details).
Without skill points, skills are all balanced (roughly), and they scale up with Clvl, attributes, equipment, and more. This makes many more skills viable for the end game, and allows much more skill experimentation.
The entire attribute system was overhauled in Diablo III. See the attributes article for more details, but in terms of spell damage, the key information is that spell damage is calculated much like weapon damage, with a base value that is modified (increased) by attributes, equipment bonuses, weapon stats, and more. This entirely changes the build dynamics for casters, who could use attribute points purely for vitality or equipment requirements in Diablo II.
Spell damage is calculated much like melee weapon or ranged attack damage was calculated in Diablo II. This is a big change, since in Diablo 2 spell damage was determined chiefly by equipment bonuses, skill points (including synergies), and Clvl. In Diablo III players with caster characters are as dependent upon good equipment, especially weapons, as combat characters.
In Diablo III, casting speed is determined not by an inherent character property, but by the item equipped. Faster hitting weapons allow for faster spell casting. Furthermore, DoT effects have their "tick rate" set by the weapon that inflicts the damage. This also makes faster attack bonuses as useful for mages as for combat characters, and provides a built in advantage to lighter damage/faster weapons, and a penalty to higher damage/slower weapons.
+% Spell Damage Affixes
Early in Diablo III's development, the developers talked a great deal about the importance of +% spell damage modifiers. Their initial design theory was that mages would focus their equipment goals on assembling as much +% damage as possible, much as combat characters do. Over the course of the game's development, this system changed, and as of the Beta test +% spell damage equipment seems to have been removed from the game. Some may still appear at higher levels, but if so it will likely be a fairly rare modifier.
Interestingly, +% damage bonuses on weapons has been greatly diminished in importance as well, with the % bonuses quite small. For instance, numerous high level Legendary items have +30% or less bonus damage. As a result, modifiers such as +attribute bonuses are much more important and impactful on total character damage.
Spell Damage Explained
In December 2011 one of the lead developers of Diablo III's skill system, Wyatt Cheng, posted a lengthy and highly-informative write up of the current functions and issues with the skill damage system. 
First, a few qualitative comments:
Yes, your build absolutely matters. Some builds may be well designed to favor 2-Handers, and some may favor 1-handers. So the general statement of “The correct weapon depends on your build” overrides everything else I write.
Additionally though, two general rules of Diablo weapon speeds still applies – spilled damage matters, and speed matters for combat effectiveness. In WoW, you might use a 1.5 second cast spell over a 2.5 second cast spell because you simply can’t afford to stand still that extra 1 second. How many spells in WoW do you cast purely by virtue of them being instant cast or castable while moving? In Diablo, virtually every spell is working off of your weapon speed, and the need to be able to quickly move 1/10th of a second sooner sometimes matters. Spilled damage matters too. Many aspects of Diablo involve fighting hordes of small monsters. When a monster has only 150 health, who really cares that you hit it for 300 damage instead of 200 damage? Either way it’s dead. And if it’s dead with a faster weapon, that means you can get to the next monster faster. This means that a weapon with a 1.3 (attacks per second) speed that can 1-shot enemies actually has a 30% killing throughput increase over a weapon with a 1.0 (attacks per second) speed.
“But I don’t care about these edge cases”.
Well first, I would argue that in Diablo, these aren’t edge cases, they are a core part of the gameplay that come up all of the time.
HOWEVER, for those who are insistent on maximizing theoretical DPS and AP usage, we’ve decided to cover you anyways.
First, straight up. all things being equal, 2-handers do more damage than 1-handers. This is pretty obvious, but I want to confirm and validate this. The reason it’s important to call this out is that all the calculations assume “for any 2 items of the same DPS”. But that’s not a great starting point, because for any two comparable items at any given level, 2-handers do at least 15% more damage than 1-handers, and in many cases 20-25% more.
To put it another way, many of the posts in the linked thread make an assumption that two weapons are equal DPS. This assumption is flawed – what you really want to ask is “For the theoretically best 2-hander in the entire world (even though I’ll never get one)” vs. “the theoretically best 1-hander in the entire world (even though I’ll never get one)”.
However, what you choose to put in your off-hand, should you choose to wield one, matters.
First, no matter WHAT you use, you are getting a bunch of extra stats. In addition to DPS from Attack and Precision, you’re getting whatever other item stats are on your offhand. So you’re trading off a theoretical Arcane power efficiency boost for the stats of an offhand.
Furthermore, if you use a shield, you’re getting a big armor boost. If you haven’t played Diablo in a while, people easily forget, but a lot of Sorceress players used a shield in Diablo II. For some players there is a “fantasy” of “I don’t need a shield ’cause I don’t plan on getting hit”. The reality is that we don’t let you get away with that in Diablo. You get hit. We don’t have heal, tank and DPS roles in Diablo, so everybody in Diablo eventually takes damage.
Additionally, if you choose to use an orb, every wizard Orb (and witchdoctor Mojo) comes with +dmg on it. So if your mainhand does 8-10 damage at 1.4 (Attacks Per Second) speed, and your orb adds 3-4 damage, then that means you’re doing 11-14 damage at 1.4 speed. In many cases the orb + the stats on the orb completely closes the 15-25% DPS gap between 2-handers and 1-handers.
Add on top of that +dmg from your rings and amulets, and currently with internal tuning numbers (this may not be how we ship), but 1-handers + offhand out-DPS 2-handers almost all of the time. It takes a luckily rolled 2-hander to out-DPS most 1-hand + offhand setups if you can also spare some +dmg on your rings and amulets.
Finally, fast weapons are better at fishing for procs. Diablo has lots of very cool proc effects. Chance to gain Arcane power on critical hit. Chance to summon a fetish on spell cast passive on the Witchdoctor. Faster and more frequent casts means more opportunities to fish for procs.
Where does that leave us? Well currently for internal testing of Inferno mode, 1H+Orb is overwhelmingly better. You get more damage, better mobility, more stats, and more proc fishing. 2H gets you better Arcane Power efficiency. If anything I’m currently worried that 2H is too weak. The most likely solution on this front will be to reduce the amount of +dmg found on rings and amulets to reel in the damage advantage of 1H+Orb. However, the value of Arcane Power efficiency varies the more you have to run and move. If you’re running and moving constantly, then AP efficiency lets you drop big bombs like meteor and Hydra when you finally get to stand still. Since the amount of running and moving varies from situation to situation, I’m actually fairly happy with where things are.
On the topic of Disintegrate and Arcane Torrent, we’re actually currently considering a change to make them drain AP faster, to match the drain rate on all other skills. I haven’t decided yet whether it’s better for the game for the philosophy on AP to remain consistent across all skills, or if it’s cool that 2 skills break the rule. (Ray of Frost breaks it too BTW). I am currently leaning towards having it drain AP faster because although I absolutely LOVE having different skills for different builds, I also think that when it comes to your resource system, having some standardized themes is what makes it possible to design all the OTHER skills/systems that break those rules. For example, standardized “fast weapons drain AP faster” is also what let’s us do “Arcane power on Crit” that let’s you feel like you’re cheating with all the skills.
On a side note, I don’t think you actually get any potential DPS increase by being able to squeeze in an extra cast in between casts of meteor because the time of 4.8 seconds is the amount of time it takes to regenerate back the AP cost of Meteor, but it’s not like any additional AP just ‘disappeared”, you still have it. A 65 AP cost spell gives you a 35 AP buffer to pool over multiple casts. So let’s say that you figure out your theoretical breakpoint is “6.9 electrocutes in between every meteor” and you think you’re going to get an extra cast if you can get that to “7 electrocutes between every meteor”. That’s not really correct because 6.9 actually plays out as “Meteor // 6 Electrocutes"... repeated 9 times followed by “Meteor-7 electrocutes”.
To be even more pedantic, (but I know you theorycrafting folks geek out on this stuff anyways), this has never actually come up as a balancing issue in any of our internal tests because a Wizard never stands still shooting long enough for this to ever actually matter. The truth is that you’re going to weave in other skills, or have to spend time moving, or whatever else long before anything in the previous paragraph matters.-Wyatt
One type of "spell" that can't be considered a skill is something cast by a scroll. There were several scrolls found in Diablo III during development, but as of the beta test only two remain:
- The Scroll of Identify works just as it did in Diablo II. These items are much less needed in Diablo 3, since blue (magical) items do not need to be identified. Only Rares, Sets, and Legendary items require Identify scrolls.
- The Scroll of Companion is a new type of spell. These summon a small "pet" that scurries (or flaps) around the dungeon, picking up gold for you.
- The Scroll of Wealth was present during development, but was removed in 2011. Its function has been replaced by the Cauldron of Jordan.
- The Scroll of Town Portal was also present during development, but was removed since it provided an easy and exploitable escape hatch. The Stone of Recall allows quick trips to and from town, but there is a lengthy casting time, which is interrupted if the player moves, casts a spell, attacks, etc, during it.
+%Spell Damage Retrospective
As a useful memory of how important the +% spell damage modifier was set for, read this quote from Flux's post-Blizzcon 2009 Wizard write up: 
- When viewing the listed damage on each spell, it’s important to keep in mind that those are not absolute numbers, in D3. They’re much like weapon damage; modified by your wizard’s attributes, your character level, by other skills, and also by your equipment. Lots of types of items in D3 carry modifiers that boost spell damage, usually by a percentage.
- I found numerous wands, staves, and even pieces of armor with values between +5-25% spell damage, and that was just in the early stages of the game. We know nothing about higher level equipment bonuses, and it’s entirely possible that the D3 team is projecting high level characters to have +100%, +200%, or who knows, +500%, spell damage. If so they’ll be adjusting the base spell damage accordingly, which might make it look very low to our untrained eyes.
- Here’s an example from early in the game, that was mentioned in the Witch Doctor skills report. The skill tree listed damage on my level 2 Skull of Flame spell was 6-9, yet with a level 12 Witch Doctor my Inventory screen showed 19-29 for that spell. I don’t think I had more than +50% spell damage from equipment, so assuming the display values aren’t wrong (in D2’s infamous LCS style), there are considerable bonuses added to damage from character attributes, which are then further boosted by the spell damage bonuses on your equipment.
- This is all part of the D3 team’s plan to make all of the attributes useful to all characters, and to make more types of modifiers useful/essential to spell casters. We don’t know enough yet to judge how well it’s working, but it should be a substantial change from how spell damage was calculated and scaled up (or not) in D2.