Mr. Wemmick is Jaggers's clerk—sometimes. When he's the clerk, he's gruff, business-like, and "dry" (21.1), who has to remind himself that people have the "habit of shaking hands" (21.28) and who constantly insists on keeping "portable property" (24.41).
But when he's at home in Walworth, he's entirely different. He's built himself a little castle—or, rather, added some "sham" gothic windows and a teeny tiny drawbridge—where his "hard face" softens and he spends his time puttering around and fixing up his house and yard" (25.44). The difference between Business Wemmick and Home Wemmick is so great that Pip almost thinks there are "twin Wemmicks" (48.13).
Other than this funny split personality, we don't know much about Wemmick, but we don't need to. He's a way for Dickens to comment on—and make a little fun of—the Victorian idea of separate spheres, the idea that a man's home is his castle (literally, in this case), and that work and home should be kept totally separate. Obviously, Wemmick is taking it a little too far—but he makes us wonder if we should really be doing it at all.