Before answering the question in the title, we should clarify what the difference actually is. A blog, after all, is a website, and some websites that are exclusively blogs are quite successful. When it comes to launching an online presence for the first time, it is important to understand the subtle differences.
When we say “whole website”, it means your site does something beyond publishing. It may have an ecommerce shopping cart, an interactive community registration system, or any other focus. Blog hosting is best for publishing content, and since “content is king”, as they say, some websites include blogs in addition to their other services.
In my days as a web designer, I met many clients who said they wanted websites. Most of them wanted something very simple, a virtual business card that they could show to their customers and say, “Hey look, I’m technologically advanced because I have a website.” Some probably just wanted customers to stop asking them if they had one. For those people who have no intention of keeping a site up-to-date, having a blog will only make their site appear stale and old. A blog that says “Hello World!” and has a 2009 timestamp will have that effect.
For those clients who were launching online magazines, news sites, or even just sites to give them an outlet for their self expression, having a blog made sense. Those clients need blog hosting that gives them a wide range of features out of the box, while also giving them the flexibility and scalability (in case they decide to have more than a blog) that most free blog hosting services cannot provide.
A blog content management system like WordPress is an excellent choice because it is free, open source, and extensible. If you are in the market for blog hosting, you should look for a web host that will install and manage your blog software as part of your hosting package.