I think the fight was simple, to over-thrown Hashirama from Hokages seat, or maybe he just wanted to have a final spar to prove who is more worthy.
The fight is over, Madara have lost and is mortaly wounded with a sword pierced through his chest. However, Hashirama shows compassion and understanding towards Madara. They were afterall a very fierce rivals that just like Naruto
and Sasuke, can understand each other better by "exchanging fists". Hashirama thus felt sorry/pity for Madara and spared his life by using his cell regeneration jutsu on Madaras wounds. Basically, Hashiramas cells restored Madaras body.
Just remember with how much respect Madara talks each time he mentions Hashirama.
In return for his life, Hashirama told Madara to stay away from shinobi matters, that from that day on he was to be dead and never to re-emerge in Konoha ever again. Possibly, Hashirama may have used talk-no-jutsu on Madara to convince him that he would be a strong leader for Konoha and that he will ensure that peace will last.