Question

how is the home page of a browser different from the home page of a website

Answer 1

Basically by default the homepage of a browser is at most part a search engine, say it google, firefox, bing etc.

While on the other hand the homepage of website is like the MAIN PAGE which then divides into sub sections like topic etc if its a blog etc

hope it solves your query, if you still find any doubt google is always there for your need ; )