They can use it, but if the random roll falls on a spell, it fails
The rule for casting multiple spells in one turn says:
You must use a bonus action on your turn to cast the spell, provided that you haven't already taken a bonus action this turn. You can't cast another spell during the same turn, except for a cantrip with a casting time of 1 action.
So after casting a spell as a bonus action, there is nothing stopping you from using the wand as an action, but you cannot cast any spells above cantrip level on this turn.
Casting a spell from a wand is still casting a spell and plays by the same rules
While holding it, you can use an action to expend 1 of its charges and choose a target [...] Roll d100 and consult the following table to discover what happens.
If the effect causes you to cast a spell from the wand, the spell's
save DC is 15.
The wand of wonder, like all wands, counts as you casting a spell. It is even in the description for the item. All the spell effects say:
You cast [Spell].
There is nothing that would excuse this casting from the rule that says:
You can't cast another spell during the same turn, except for a cantrip with a casting time of 1 action.
And none of the spells contained withing the Wand of Wonder are cantrips.
Thus, if they used the wand after casting a bonus action spell and a spell effect is rolled, the spell would fail because it is not a cantrip.
Any of the non-spell effects would happen normally
There are no restrictions on the activation of other features of magic items though so it would not make any sense to disallow the use of the magic item entirely in this case.
If they cast a spell beforehand though, they will have to accept that there is a risk of them wasting their action with the Wand of Wonder.
Not if the spell being cast is 1st level or higher.
Per the spellcasting rules (p202 PHB), if a spell is cast using a bonus action, only a cantrip may be cast using your action.
BONUS ACTION
A spell cast with a bonus action is especially swift. You must use a bonus action on your turn to cast the spell, provided you haven't already taken a bonus action this turn. You can't cast another spell during the same turn, expect for a cantrip with a casting time of 1 action.
This section makes no reference to expending spell slots, only the casting of spells. The wands should still be covered by this general rule.
But are YOU casting a spell when you use the wand, or is the wand? The Wand of Detect Magic states
While holding it, you can expend one charge as an action to cast the Detect Magic spell from it.
Here, a charge is analogous to a spell slot. You use a finite resource and as a result you cast a spell. The "from it" part could sound like the wand is doing the casting, however this is probably to signify that the wand itself is being used a focus for the spellcasting. The spellcaster is still the one casting spells, not their focus.
Under the Magic Item section of the DMG, on p141 there is a section regarding items that when activated casts spells.
SPELLS
Some magic items allow the user to cast a spell from the item.
This further leads credence to the point that the user is casting the spell as opposed to the item.
Also on p141 of the DMG, under Activating an Item.
If an item requires an action to activate, that action isn't a feature of the Use an Item action...
Activating one of the wands doesn't come under the Use an Item action.
It appears that the general rule for bonus action spell casting still holds true for using magic wands. As the user is the one casting the spell, they are limited to the use of wands that produce cantrip spells, if they have already cast a spell with their bonus action.
As an aside, allowing the use of wands in this manner would definitely skew the action economy in favour of the spellcaster. Wands are typically an inflexible but rechargeable source of extra spell slots, which can be quite powerful. They free up prepared spell lists and set aside additional slots for use. Allowing them to circumvent the standard spellcasting restrictions as well would make them even more powerful. The first cheese tactic I can think of is the spellcaster getting surrounded in melee combat, then Misty Steping away and Fire Balling the crowd of mobs before they have a chance to react.
Best Answer
Description vs Mechanics
There is a difference between the phrase
and
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:
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.