So, let us start with where 4e is today.
Today, 4e is mostly balanced for 2-3 encounters per extended rest. You can certainly go 2, and can probably go 3, before you need to take a break. You also have a rough parity in your powers: a martial exploit of level X is roughly the same as an arcane spell of level X. They are different, but they do similar damage and have similar effects.
Given that we have a good measure of how long it takes a monster to die, we know how many hits a normal 4e character makes over the course of a day with 3 encounters in it: 4-8 per monster (lets parse that down the middle and say 6), in each of 3 combats, means about 18 hits between extended rests (from 12 to 24, depending on how efficient you are at killin').
18 hits is an important number, as we shall see…
Show Me The Pillars
In 4e, all of these rolls are done in the context of combat. In 5e, with a recognition that combat is only one part of the game, we look at adventure design, and we look at it in the context of the three main things that adventurers do: combat, interaction, and exploration. If we say that an adventuring day should be roughly evenly divided between the three activities, we come up with about 4-8 hits per activity per extended rest for each character. Or, to simplify: 6 successes (hits) each for each character in combat, interaction, and exploration.
If we say that a standard "challenge" takes 6 successes to overcome, then we have a way to draw a parallel between a goblin and a locked door and a stubborn guard: each are worth 6 successes. Monsters and traps and NPC's can offer the same challenge regardless of the pillar they threaten.
A day's challenges then look like this:
6 hits = 1 monster = 1 Challenge = 6 successes; 1 monster/party member = 1 challenge/party member = 6 successes/party member; 3 monsters/party members/day = 3 challenges/party member/day = 18 successes/party member/day.
What's the Difference Between Spells and Swords?
4e shows that it is possible for the differences between spells and swords to be very minimal. Essentially, they can be identical for all practical purposes. Use a spell or hit with a sword, and it's generally a wash.
So that math demonstrates that a hit is a hit is a hit. A blast of fire, or a hit with a sword. A convincing argument (Diplomacy check) or mind-altering magic (Charm Person). A clever lockpick (Thievery check) or a magic force (Knock). These can all be modeled with the exact same mechanic.
That is, however, kind of boring.
So, how Vancian magic changes this equation is by giving spellcasters a few "slots" that can be filled with spells that they prepare. The spell list that they prepare from might be quite large, but they must choose a subset of those spells to use each day, and once a "slot" is spent, it is only recovered when you take a long rest to prepare your spells again.
The logic runs like this: a fighter can swing his sword over and over and over again. A wizard can only use a spell once, and then it is gone. In exchange for that limitation, the wizard's spells are more powerful.
4e shows that this math can work, on a basic level. If a hit is a hit is a hit, then you can give up a hit on one round to get two hits on your next round. Double the power, but half the chances to use it. A wizard swings drastically in power over the course of a day, but a fighter or rogue stays relatively on par. This is a dramatically different psychological experience, and that difference is worth preserving: having to think about a challenge strategically, over the long haul, helps one play the role of a brilliant-minded wizard, doling out a few very powerful effects when needed most.
This distinguishes the Wizard from the Warrior in terms of game mechanics. The wizard's spells are a powerful, limited resource, that must be applied strategically and intelligently to their best effect. A warrior's attacks are less powerful individually, but can be done repeatedly.
Now that we've made locked doors and goblins and NPC's roughly comparable, we can also compare that "spike" across different pillars. If Acid Arrow is worth two hits, but can only be done three times in a combat, we're balanced. If Charm Person is worth two Diplomacy checks, but can only be done three times in an encounter with an NPC, then we're also balanced.
Of course, the spike is usually much more dramatic than that. We do have a metric for comparing that dramatic spike: our "number of hits to kill a monster" metric.
Do One Thing, Do It Well?
Lets start with an extreme position: fighters have only at-will abilities, and wizards have only one daily ability. Can this be balanced?
Yes. Over the course of a day, a 4e fighter makes, say, 18 attacks. The wizard makes one attack that deals eighteen times the damage that each fighter attack does. This is a form of balance. An extreme form, but still, it illustrates our purposes.
This is applicable to all three pillars. The wizard might, instead, use one spell that is the effect of eighteen Diplomacy checks. Or eighteen Stealth checks. And that's all they do. After that, nada.
A Bit More Practical…
Okay, more than likely you're going to want to give wizards more than one thing to do in a day. And probably, you want to give fighters some spike potential (but that's a different conversation). So you just need to figure out how you want to split up. One way to do it might be to let wizards prepare three spells each day (so that each spell is worth about 6 successes). This lets a wizard make a few strategic choices: what spells do you prepare? What challenges are you likely to face? Do I think we're going to face more combat (and so I'll double up on combat magic)? Do I think we'll need more interaction spells? It also lets a wizard be broad: they can memorize one spell per pillar, and be capable in each situation.
So we have that as a baseline: a wizard can prepare three spells each day. Each spell is worth about 6 successes (6 normal attacks, 6 Diplomacy checks, or 6 Thievery checks).
Controlling the Recharge
So, an immediate strategy emerges from this: maximize the amount of time the wizard can unleash their powerful spells. "Go Nova, then Go Home." This is the "15-minute adventuring day" problem. It can be controlled in a few different ways, and I talk about them in the What Recharging Looks Like musing.
A Note On Adventure-Based Design
Someone looking at this might immediately say, "Hey! SIX TIMES the power of a fighter is HUGE! It can kill monsters all at once!"
That's also fine.
In adventure-based design, each encounter need not be micro-managed. You can have a spell (or an Assassin's death attack!) that kills a creature outright, or that makes a particular encounter a cakewalk. This is because the adventure, as noted above, cannot be parceled out into individual encounters. If the party is to accomplish their goal, get the treasure, and get XP, they must complete the adventure before taking an extended rest. If they take a rest in the middle of the adventure, they leave it uncompleted. This means that even if a wizard dominates the first encounter or two, after that, they still need the rest of the party to do everything else for them.
Optional: The Safety Cast
In order to give wizards a bit less encounter dominance, you might want to consider an optional rule that gives them a spell failure chance if they take damage on their turn. If the wizard took damage between their last turn, and this turn, they must make a Constitution check to successfully cast the spell. This encourages the wizards to keep their allies around even in encounters that they can win handily, simply so that they are ABLE to get their spells off.
This can make the rest of the party seem a little bit like "wizard bodyguards," however, at least for a few encounters, so it might not be for everyone. It tends to make the wizard the focus of any encounter in which they have spells, which isn't usually a desired outcome.
The "three spells per day" model should stay fairly static. That is, as the wizard gains levels, they don't get to cast more spells. They do get to cast more POWERFUL spells, filling those slots with increased power, but their capacity generally doesn't increase.
It could, if you allowed the entire party's capacity to increase. For instance, if a typical creature took 6 hits at 1st level, but only 3 hits at 10th level, that's doubling the power. You could then let a wizard use two spells. The important thing is that they all increase equally. This is actually similar to how early editions scaled, but at high levels, this becomes very swingy and sudden, so if you do this, you probably want to be very careful and limited about how and when you do it.
Other Spellcasting Systems
The Vancian system, while it can be balanced, certainly isn't to everyone's playstyle. Which is why it's important to have multiple classes with varying degrees of spell replenishment.
A warlock, for instance, might be more like "the fighter-style wizard," with reliable spells of less potency that it can use over and over again — essentially, a class made up of at-will powers. This would be the spellcaster for the "pew pew pew" playstyle: it shoots magic at things until they die. A sorcerer might be something of a middle ground: fewer spells that recharge faster, so instead of 3 spells per day, they get 1 spell per encounter.
Meanwhile, you might tap a class like the Barbarian to be more "the wizard-style fighter," with Rage enabling them to perform TREMENDOUS feats of strength and willpower (six times the power of a normal fighter! six times the result from a Jump check! six times the ability to ignore attempts to persuade them!), but only a few times in a day.
There's a lot of variation possible within these bounds.