As nouns the difference between grandmother and baby
is that
grandmother is a mother of someone's parent while
baby is a very young human, particularly from birth to a couple of years old or until walking is fully mastered.
As an adjective baby is
of a child: very young; of the age when he or she would be termed a baby or infant.
As a verb baby is
{{cx|transitive|lang=en}} To coddle; to pamper somebody like an infant.