Fully anonymous
Your name never shows up on the story viewer list and no notification is sent. The person you watch has no idea you were ever there.
Why use a Snapchat story viewer
Every time you open a story in the app, Snapchat adds your username to a list the poster can pull up by swiping on their own snap. It shows who watched and in what order, and it stays readable for as long as the snap is alive.
That list cannot be edited. Once your name is on it there is no setting that takes it off, and opening the snap again only moves you back to the top of the order. Deleting your own account is the only thing that clears it, which is a lot to give up over one story.
Watching in a browser steps around the whole thing. No account is signed in, so there is no username to attach to the view, nothing is added to the list and no notification goes out. The poster sees exactly what they would have seen if nobody had opened it.
Who uses a Snapchat story viewer
The largest group has no Snapchat account at all. Someone shares a public profile link, and the app asks for an install, a phone number and a sign up before it will show a single snap. A browser turns that into one step with nothing to commit to.
Then there are people who work with the platform. Social media managers check what a competing brand posted without the company account appearing as a viewer, and creators pull reference from public accounts in their niche before the day resets and the stories are gone.
The rest are ordinary users being careful. Parents look over the public profiles their kid follows without setting up a second account, and plenty of people just do not want a passing bit of curiosity turning into a notification that somebody else gets to read. If it is only the profile photo you are after, the Snapchat profile picture viewer pulls that on its own.