How does this work sense the person with snatch arrow can immediately throw the weapon back or keep it for later?
When they throw it back does it just vanish?
or
Can the wizard get hit by his own weapon?
Snatch Arrow: Thrown weapons can immediately be thrown back as an attack against the original attacker (even though it isn't your turn) or kept for later use.
Hand of Apprentice: As a standard action, you can make a single attack using a melee weapon at a range of 30 feet. This attack is treated as a ranged attack with a thrown weapon.
I don't read Hand of Apprentice as being some nebulous pseudo-weapon. It's the actual weapon in your hand. A real physical weapon. It's just a magically-assisted throw of your weapon.
Nothing would vanish, because nothing is conjured. It's really no different than any other thrown weapon. If a rogue throws a dagger, would Snatch Arrow allow it to be returned aggressively?
It's a magically assisted throw of your weapon followed by the weapon instantly returning to the caster's hand:
You cause your melee weapon to fly from your grasp and strike a foe before instantly returning to you. As a standard action, you can make a single attack using a melee weapon at a range of 30 feet. This attack is treated as a ranged attack with a thrown weapon, except that you add your Intelligence modifier on the attack roll instead of your Dexterity modifier (damage still relies on Strength). This ability cannot be used to perform a combat maneuver. You can use this ability a number of times per day equal to 3 + your Intelligence modifier.
It's unclear whether "instantly" means "before the snatcher can throw the weapon back".
The snatcher should not be able to keep it for later, as the item will simply vanish and return to the caster's hand.
Oh, i forgot about that. That does make it trickier. Does it return via teleport or more like a returning weapon then? I'm seeing this as a specialized version of Mage Hand, in which case I'd let the Snatcher have a go. Maybe an opposed check to wrest control?