The monk's Deflect Missiles feature description says:
Starting at 3rd level, you can use your reaction to deflect or catch
the missile when you are hit by a ranged weapon attack. When you do
so, the damage you take from the attack is reduced by 1d10 + your
Dexterity modifier + your monk level.
If you reduce the damage to 0, you can catch the missile if it is
small enough for you to hold in one hand and you have at least one
hand free. […]
The description of the Gloves of Missile Snaring says:
These gloves seem to almost meld into your hands when you don them.
When a ranged weapon attack hits you while you're wearing them, you
can use your reaction to reduce the damage by 1d10 + your Dexterity
modifier, provided that you have a free hand. If you reduce the damage
to 0, you can catch the missile if it is small enough for you to hold
in that hand.
Does a monk's Deflect Missiles ability work "in addition" with the Gloves of Missile Snaring, or must you use one or the other?
According to the description of reactions (PHB 190):
Both use your reaction. As you only have 1 reaction per round, you must choose which of the two you would like to activate.