Jacobson rings were introduced independently by Wolfgang Krull (1951, 1952), who named them after Nathan Jacobson because of their relation to Jacobson radicals, and by Oscar Goldman (1951), who named them Hilbert rings after David Hilbert because of their relation to Hilbert's Nullstellensatz.
Hilbert's Nullstellensatz of algebraic geometry is a special case of the statement that the polynomial ring in finitely many variables over a field is a Hilbert ring.
A general form of the Nullstellensatz states that if R is a Jacobson ring, then so is any finitely generated R-algebra S. Moreover, the pullback of any maximal ideal J of S is a maximal ideal I of R, and S/J is a finite extension of the field R/I.
This explains why for algebraic varieties over fields it is often sufficient to work with the maximal ideals rather than with all prime ideals, as was done before the introduction of schemes.
For more general rings such as local rings, it is no longer true that morphisms of rings induce morphisms of the maximal spectra, and the use of prime ideals rather than maximal ideals gives a cleaner theory.