Schlessinger's criterion is a criterion for the pro-representability of a functor. This is the same thing is getting something like an object on the formal scheme $\mathfrak V$. Grothendieck's result tells us that this object comes from an actual object on $V$. Usually this is the step after pro-representability to get actual representability (and is usually formulated as saying that the formal deformation is "effective", a property that goes beyond pro-representability). This is precisely the setup of Artin's criteria for representability by an algebraic space.
Schlessinger's criterion is a criterion for the pro-representability of a functor. This is the same thing is getting something like an object on the formal scheme $\mathfrak V$. Grothendieck's result tells us that this object comes from an actual object on $V$. Usually this is the step after pro-representability to get actual representability (and is usually formulated as saying that the formal deformation is "effective", a property that goes beyond pro-representability). This is precisely the setup of Artin's criteria for representability by an algebraic.