Sir Walter approves Elizabeth's friendship with Mrs. Clay in Persuasion, but not Anne's friendship with Mrs. Smith. Why?
Mrs. Clay is wily and sufficiently flattering to Sir Walter that she deceives him (and Elizabeth) into thinking she is trustworthy as well as worthy of the family's attention. She is also the daughter of Mr. Shepherd, family advisor to Sir Walter. Sir Walter is unable to see the deceptive intentions of Mrs. Clay; she "persuades" both Elizabeth and Sir Walter that she is more than what she truly is. It is only Anne and Lady Russell who see through her.
Keep in mind, too, that Elizabeth continually tries to reflect the superficiality and class conscientiousness that define Sir Walter. Read the e-notes section (cited below) on class consciouness as one of the book's important themes. Although both Mrs. Clay and Mrs. Smith are of a lower class, Mrs. Smith does not attempt to appear to be of a higher class than she is. It is Mrs. Clay's deceptiveness and Elizabeth's adherence to what Sir Walter foolishly believes that leads him to accept her and later to reject Mrs. Smith.
Mrs. Smith, however, in unpretentious and honest. She has fallen on hard times and, in Sir Walter's view, is not worthy to be Anne's friend. Anne, who sees past such superficiality, sees the worth in her good friend.