Both live on Account.
Your name is used in the emails we send you, and it's the fallback From name on the emails your quizzes send to your leads when you haven't set a display name.
Brand or display name is optional. If you set it, quiz takers see it on the email step of your quiz — "Your email goes to Baker Studio, who made this quiz" — and it's the From name on the result email and any follow-ups. A brand or business name usually works better than a personal one. If both names are blank, the quiz title is used as the From name.
Replies to those emails come to your sign-in email address.
Leave it blank and the email step shows nothing: the line simply reads "Your email goes to the person who made this quiz". We never fall back to your name, your handle or your email address there.
You can change either at any time; it applies to new visitors straight away.
Sign-in details. Account → Sign-in details is where you change the email address and password you sign in with. Changing your email sends a confirmation link to the new address; nothing changes until you click it.