Rebecca is totally "horrified" that Ann Putnam would send her one surviving daughter, Ruth, to conjure the spirits of her other dead children.  However, Mrs. Putnam shouts back, "Let God blame me, not you, not you, Rebecca!  I'll not have you judging me any more," then immediately asks Reverend Hale, "Is it a natural work to lose seven children before they live a day?"  Therefore, it seems as though Ann feels that Rebecca has judged her in the past, and the fact that she links that judgment to the deaths of her children demonstrates that Ann may feel that Rebecca has judged her for those deaths (although, given Rebecca's pained response, that seems unlikely).  

When Rebecca had first entered Betty's room, her presence alone was enough to quiet the whimpering child, and Mr. Putnam asks if she'd be willing to visit their daughter too.  However, Rebecca says that she believes the child will wake when she is ready, that she'll eat when she becomes hungry enough.  For every emotional claim Ann makes, Rebecca has a measured, calm response, and she refers to her success as a mother and grandmother to bolster her credibility in this matter.  If we consider the way we might feel in a similar situation-- all our children dead, but for one living child suffering from an unknown cause-- Rebecca's reason could be quite upsetting.  The Putnams want an investigation into witchcraft while Rebecca encourages them to seek for the cause of the children's sickness within themselves.  The suggestion that she might be to blame for her children's sad fates pushes Ann over the edge.

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.