The most obvious difference is that Piaget's stages are hierarchical, while Vygotsky's are not. This means that one of Piaget's stages must be "completed" before moving on to the next; Vygotsky's theory does not depend on time. Also, Piaget's stages imply that children cognitively develop on their own. That is, a child completes one of Piaget's stages on his own, without the help of outside aid. For example, during the sensorimotor stage, a child understands that an object exists even when it is out of sight; no social interaction is needed for a child to understand that. On the other hand, Vygotsky's theory concentrates more on the social interactions and aide given to a child while he develops. For example, a child will learn by interacting and communicating with older peers and adults who are more knowledgeable. When teachers support students as they learn, scaffolding takes place, which is another example of Vygotsky's social interactions.
Piaget and Vygotsky contributed heavily toward the field of child development. Even though their theories are fundamentally different, teachers and parents can incorporate both as they help their students and children learn and grow.