The character of John Wemmick is recognized by critics as one of Charles Dickens's most successful split characters. Some even term him Dickens's "most modern man." For, as he tells Pip in Chapter 25,
The office is one thing, and private life is another. When I go into the office, I leave the Castle behind me.
So, when Wemmick speaks to Pip at Walworth, he talks to him as a friend. But, when he talks with him in Mr. Jaggers's office, Wemmick is the clerk of Mr. Jaggers and does not overstep his position or even indicate that he has any personality or warmth. Much like Mr. Lorry of A Tale of Two Cities, he is strictly "a man of business" in London.
Perhaps, by protraying Wemmick as such a totally different person at Walworth, which is outside London, Dickens wishes to underscore Pip's first impression of the city as "ugly, crooked, narrow, and dirty" (Chapter 19).
Wemmick lives in two separate worlds and he tells Pip that he wants to keep these worlds separated from each other. In Jaggers' office, where he works, he is just like Jaggers. He is business-like and agrees with whatever Jaggers says. At home, however, he is a different man and often does not agree with Jaggers' way of doing things. When he gives Pip advice in one place that differs from what he has said elsewhere, he usually winks or makes some other sort of comment to remind Pip that this is the "office" Wemmick talking or this is the "home" Wemmick talking.
Critics believe that Wemmick's split personality can mean many things in this novel, one of which is the theme of appearance vs reality in the novel. Check it out here on eNotes.