One of the effects of the Wand of Wonder is:

You enlarge the target as if you had cast enlarge/reduce. If the target can't be affected by the spell, or if you didn't target a creature, you become the target.

There are multiple effects with a similar wording. There are also many effects which say instead that "you cast [spell]".

Mechanically, what are the differences between the two wordings (if any)? What exact aspects of the spellcasting process/effects are different between the two?

Description vs Mechanics

There is a difference between the phrase

as if you had cast

and

you cast [spell name]

The former is using an existing mechanic as a description for an effect, while the latter is stating you are using that mechanic itself with all of the spellcasting requirements detailed in Chapter 11 of the PHB (unless it bypasses those, but that's specific over general.)

If WoTC had meant for them to be equivalent, they'd have used identical language. By utilizing them differently in different places, there is an effective difference.

When as if you had cast is used, it is providing the effects without requiring any of the casting rules. Simply by saying as if you are purposefully differentiating from you did.

Note also that there is no opportunity for the creature to save against this. It simply states that:

You enlarge the target...

No save also suggests that this is outside the normal spellcasting mechanics. It's just something that happens and for ease of use it works like an existing mechanic.

The mechanical differences

Because both of these are spell effects but originate differently, there are different mechanics to apply to them.

You Cast

In this case, you are casting the spell. You will need to follow all of the general spellcasting rules of Chapter 11 (components, concentration, targeting, etc.) unless the specific description of the item you are using overrides them. Because you are casting, this also becomes subject to a potential counterspell.

As if

In this case, there is still a spell effect, but it's outside of the general spellcasting rules. You will still follow the description in terms of duration, but this isn't a normal cast spell. Because you haven't actually cast the spell, there are also no concentration mechanics if applicable. The effect is just ongoing for the duration because you haven't actually cast the spell. If you didn't cast it, you don't need to concentrate on it. Think of this like someone casting a buff on you. It's an effect provided by a third party source that you don't have control over.

However, it is still a magical spell effect and would be affected by other abilities and things that would normally work. There is still a reference to the spell and that it works as if you cast it, but you didn't cast it and it's just a spell effect created by something external. This also means that the effect is not available for counterspell, as there is no actual casting. But it is still a spell effect (because the description is referencing that spell), so you could use dispel magic or some other spell/ability that interacts with spell effects or magical effects.

You shrink {enlarge} yourself as if you had cast enlarge/reduce on yourself.

It's a statement linked to a particular spell¹

I think that you are over-complicating this semantic / usage difference.

The only two "as if you had" statements have to do with the same spell's effect (Enlarge Reduce, 2d level, transmutation) which is to either shrink or enlarge (caster's choice) a target. The wand casts the spell (not the caster) thus "wand casts it as if you had" to create the magical effect.

But wait, the wand always chooses! Roll d100 for results

For this particular spell, when the caster chooses themselves the caster won't get a saving throw since a saving throw is for an unwilling creature.

If the target is unwilling, it can make a Constitution saving throw¹.

(I can see different DM's choosing to rule differently on that due to the overarching "spell save DC 15 versus effects" ... but more below).

• Compare this to the 98-00 roll, where the save versus petrified is spelled out. So too the 91-95 roll, and the save versus being blinded.

Enlarge or shrink, with no save, for the wand's wielder

The wand splits that spell's magical effect into two discrete effects so that it either enlarges or shrinks the caster (or target) without the caster getting to choose. When you (the caster casts that spell) the caster chooses, but with the wand casting "as if you had" the wand / die roll chooses. Usually, when a caster casts Enlarge Reduce on themselves, they don't role a saving throw against that spell because they want to enlarge or reduce: they are not an unwilling target¹.

"As if you had" means that the wielder grows or shrinks if the target can't be affected by the spell - that spell effect backfires on the wand's wielder.

The general case for this wand is:

"If the effect causes you to cast a spell from the wand, the spell’s save DC is 15."

This wording also avoids a fizzle.

Since the wand has variable results, the wielder could be trying to cast a spell at something that is not an eligible target creature. (Even though the wand's results are always driven by a die roll ...) In the case of this roll, 50-53 on the d100 roll (or 66-69 for shrink), in a situation where the target is an eligible object or creature, the spell would otherwise fizzle if the wand description says "you cast Enlarge Reduce."

This change in language telegraphs an intent: if what you are aiming at can't be enlarged or reduced, then the spell is cast on the wand's wielder.

• Note: this wand and the Wild Magic Sorcerer subclass's Wild Magic feature have some very similar results waiting for a triggering of the effect. If one looks at this item and attempts to capture the spirit of the item, and the chaotic nature of the outcomes, rather than beating the semantics out of it, I think the interpretation becomes clearer. Your grow or shrink, Alice. (And if a white rabbit shows up, so much the better!)

Each other casting listed will otherwise adhere to casting rules, since "you cast the spell, and save DC is 15" is the overriding guidance. This roll, 50-53, and the 66-69 roll, are uniquely tied to Enlarge/Reduce and are presented differently than all of the others.

¹Enlarge Reduce spell text excerpt

You cause a creature or an object you can see within range to grow
larger or smaller for the Duration. Choose either a creature or an
object that is neither worn nor carried. If the target is unwilling,
it can make a Constitution saving throw. On a success, the spell has
no effect.

If the target is a creature, everything it is wearing and carrying
changes size with it. Any item dropped by an affected creature returns
to normal size at once.