Lev Vygotski was a socialist and Jean Piaget was a Swiss individualist and essentially these schools of thought explain their approaches.
Whereas Piaget sees child development a residing in the individual--the 'child as a scientist' exploring the world and driving development by experimentation--adaption and accomodation. Vygotski saw the child as a social being and development was sociallly orientated. Thus the principles of scaffolding-supporting the child until it can operate alone, zone of proximal dvelopment--knowing the limits of a childs growing knowledge and how much further the child can go at that particular time (their potential). Another essential point Vygotski made was that the best teacher of a concept was a peer who had recently grasped a concept. The theme of mediation is clear here. The next important point is his work on thought and language. He believed that action/behaviour drove external speech which developed into internal speech but the two speech modes are qualitativly different--and this internal speech developed into thought. LV also saw play as being crucial in the development of abstract thinking.
His book Thought and Language is still in print and very readable (see also the work of A Luria).