Even though Mrs. Slade and Mrs. Ansley are competitive in their friendship, there is a simple consideration here: they have each other after all these years to keep each other company after both husbands have passed away. Two bickering old ladies probably have a lot of fun, deep down, arguing about their collective memories.
At least they have their shared memories to keep them company in their old age. Competition is not a bad thing, in fact in the case of Mrs. Slade and Mrs. Ansley it has given meaning to their lives and depth to their old age.
Many relationships between friends have a competitive edge to them. Think about it--two athletes, two beauty queens, to honor students--they are in friendly competition daily for the same position on the team, the same crown in a pageant, the grade point average which will land them the title of valedictorian. These friendly competitions can encourage a higher level of performance and perhaps give the "prize" a bit more meaning and sense of accomplishment.
However, Mrs. Slade and Mrs. Ansley would never choose to be friends or even acquaintances if given the option. They have been forced into the relationship they have based on social position by marriage and traveling in the same circles. They competed for the love of the same man which can never come to any good. Resentment and bitterness are the only results one can expect from such a competition. If in fact, the competition continues throughout their lives, Mrs. Slade most definitely loses with Mrs. Ansley's slam-dunk news. Perhaps she revels in being the thorn under her "friend's" skin as often as she can.
Ironically, the women's daughters seem to genuinely be friends.