Watson includes a lot of dialogue, so the tone of his narration often shifts whenever he's talking to someone new. However, there's a distinctive feel to Watson's internal monologues and descriptions. Take, for example, this scene from his first night in Baskerville Hall:
Far away a chiming clock struck out the quarters of the hours, but otherwise a deathly silence lay upon the old house. And then suddenly, in the very dead of the night, there came a sound to my ears, clear, resonant, and unmistakable. It was the sob of a woman, the muffled, strangling gasp of one who is torn by an uncontrollable sorrow. (6.72)
Watson is fascinated by all of the little details of this scene, by the "chiming clock" and the particular "muffled, strangling gasp" of the unknown woman's sobs. He's got a great eye (or ear) for specifics, which always gives us the impression—even in less emotional descriptions than this one—that Watson is intensely interested in whatever he's reporting to us.
Check out the way that Watson presents this mysterious woman's tears. He emphasizes that "a deathly silence lay upon the old house," when "in the very dead of night" he hears sobbing. This repetition of "dead" not only gives us a sense of how still and silent the house is (answer: extremely), but it also gives a mournful, sad feeling to Watson's description of the crying he hears. Watson's choice of language here conveys a sense of emotion as well as fact, which helps capture our interest as readers. In fact, all of Watson's descriptions tend to emphasize how they make him feel, which gives us a sense not only of the scene, but also of Watson's personality.