Sir Walter Elliot has had a long and steady friendship with Lady Russell that dates back to his wife's lifetime. Indeed, Lady Russell was the particular friend of Lady Elliott and through that friendship was introduced to Sir Walter. It is this association that plays a part in how he views his friendship with Lady Russell.
Firstly, he respects and admires her for having been the friend of his good and sensible wife. Secondly, when he lost his wife, Lady Russell stepped up to fill the gap of level-headed adviser. Thirdly, Sir Walter was determined never to remarry (after having been rejected once or twice) and so leave his estate undivided and intact for his daughters, especially Elizabeth:
Sir Walter, ... (having met with one or two private disappointments in very unreasonable applications), prided himself on remaining single for his dear daughters' sake.
Therefore, though society thought it an inevitability, Sir Walter kept--and was encouraged to keep--his emotional distance from Lady Russell, with both parties acting for the sake of his daughters.
Thirteen years had passed away since Lady Elliot's death, and they were still near neighbours and intimate friends, and one remained a widower, the other a widow.
When Sir Walter associates with Mrs. Clay, circumstances are different altogether (and he has forgotten his one or two early rejections). His two still unwed daughters are situated in a new social circle and may have renewed prospects for marriage offers, perhaps even from the coveted cousin William Elliot. Sir Walter has all of Lady Russell's good counsel no matter whatever else occurs because she is and will remain faithful in discharging her obligation to fulfill Lady Elliot's dying request to look after them all.
In addition, Mrs. Clay is a very different sort of woman from Lady Russell. For one thing, she can talk such silliness as carrying on a discussion about the efficacious (good) effects of Gowland face lotion, a favorite topic of Sir Walter's. She is a flatterer and can make herself agreeable by flattering Sir Walter's vanity. Furthermore, she is very interested in receiving a marriage offer that she would not reject--unless someone better asks her first.
In summary, his two friendships with Lady Russell and Mrs. Clay are seen by Sir Walter as being very different because (1) Lady Russell is superior to him and advising him while Mrs. Clay is inferior and flattering him; (2) Lady Russell is sensible and tries to make him be sensible while Mrs. Clay is as silly (though conniving) as he is; (3) Lady Russell isn't interested in marriage to him while Mrs. Clay decidedly is interested.
There was one point which Anne, ... would have been more thankful to ascertain ... which was [of] her father's not being in love with Mrs Clay; and she was very far from easy about it ....