The conflict between Rebecca and Mrs. Putnam arises from Mrs. Putnam's anger with Rebecca. Part of this conflict emerged from the conflict between Rebecca's husband, Francis Nurse, and Mr. Putnam. Francis had rented a portion of land and over time, he paid for it and owned it. This led to land disputes with his neighbors, one of whom being Mr. Putnam.
Rebecca and Francis were also against admitting Thomas Putnam's man, Bayley, to the ministry of Salem. So, the conflict between Rebecca and Mrs. Putnam began with a family feud, largely instigated by the Putnams.
Mr. and Mrs. Putnam are also redirecting their bitterness, having lost seven babies; they've taken that grief and turned it into anger against others, in some cases against Rebecca and Francis Nurse, with whom they are already feuding. Rebecca has had eleven children so Mrs. Putnam may have been envious of her luck with surviving children, leading to more angst against her. In Act One, when Rebecca says the reason for the sick (allegedly bewitched) girls is either their own faults (the adults) or a cause attributable to God, Mrs. Putnam says to Rebecca:
You think it God's work you should never lose a child, nor grandchild either, and I bury all but one?
The Putnam's are always looking to blame others for their troubles. Given their land and social disputes with Francis and Rebecca, and the fact that they could be (and a bit understandably so) envious that Rebecca should have eleven children surviving while Mrs. Putnam loses seven out of eight, it is clear why the Putnam's have continued to view the Nurse's as enemies.