Gfrktyl

Let's open https://www.gatesnotes.com/

The title of the site is "The blog of Bill Gates"

Why "The blog of Bill Gates" and not "The Bill Gates' blog"?

Is it the possessive case or not?

*The Bill Gates’ blog

is wrong. When there is a possessive case or a possessive adjective before a noun, no article is used.

Bill Gates’ blog

is acceptable.

We now understand that “The blog of Bill Gates” and “Bill Gates’ blog” are both acceptable.

Which one is prefered?

To my taste, the possessive case is prefered when the owner description is short (a single noun or name), e.g. “Bill’s blog”; and “of + owner” is prefered when the owner is described by a long(er) name or phrase.

Also, “of + owner” puts slightly more stress on the ownership than the possessive case. As Connor Harris mentions in a comment below, this is a specific case of the general rhetorical rule that the most emphatic item in a statement should be placed at the end.

And lastly, if one wants to stress that there is only one blog of Bill Gates (and this is it), then one should use the only form that allows a definite article: “the + thing + of + owner”.

Adhemer's answer covers almost everything, but it's worth adding that "The X of Y" tends to sound grander than "Y's X". It sounds like a given title, implying that this is something special, noteworthy and unique.

"The Sceptre of the King" sounds natural (as does "The King's Sceptre"), whereas if someone talked about "The overcoat of Bob Smith", I'd think there must be some special story to this particular overcoat.

Bill Gates is famous enough that it sounds natural to call his blog "The blog of Bill Gates", but if I called mine "The blog of User 568458", it'd sound a little pretentious.

So to summarise:

• 
  X's Y e.g. Bill Gates's Blog (or Bill Gates' Blog, both are acceptable if the word ends in 's' and isn't a plural)
  
  
  
  
  • Versatile, not particularly formal
  
  
  • Could be one of many, for example this could be his general blog and there might also be "Bill Gates' music blog"
  
  
  
  


• 
  The Y of X e.g. The Blog of Bill Gates
  
  
  
  
  • Sounds formal and grand, carrying an implication that this is something the listener may have heard of
  
  
  • Implies uniqueness - that this is the only blog of Bill Gates (or, the true or definitive blog of Bill Gates)
  
  
  • Potentially ambiguous about the relationship between X and Y. Blog implies authorship, but the same construction can have other meanings, like origin ("The Holy Hand Grenade of Antioch") or subject ("The definitive encyclopedia of The Beatles")
  
  
  • May sound strange in a casual context, even if it is something grand, e.g. a palace cleaner saying "Can you pass the King's Sceptre" sounds more natural than "Can you pass the Sceptre of the King". The latter sounds like the speaker has a reason to really emphasise that this is something special ("Be more careful when cleaning the Sceptre of the King!") or, wants to show off by association ("I cleaned the Sceptre of the King today!").
  
  
  
  


• 
  The X Y - e.g. The Bill Gates Blog
  
  
  
  
  • Is even more ambiguous about authorship, but implies that this is in some way definitive. For example, I would expect a blog called "The Bill Gates blog" to be an official blog about Bill Gates, written by one or more members of his staff - and that Bill Gates would therefore not have a blog of his own that he wrote himself.
  
  
  • A Bill Gates fan might call their unofficial blog about Bill Gates "The Bill Gates blog", but it would sound pretentious unless they had some strong grounds to claim that it was the most definitive blog about Bill Gates (for example, if they were a personal confidant).
  
  
  
  

As someone with a name ending in the letter 's', I can say that I don't like the confusion caused by the possessive apostrophe added at the end in such cases.

And I don't like the .. sez sound on the end or the alternative of leaving it off.

That doesn't say anything about correct usage, but it may explain a preference.
(It may be something Bill and I have in common.)

I think that "The blog of Bill Gates" is preferable to "Bill Gates' blog" (or Bill Gates's blog") because the latter could be read as being a blog about Bill Gates, rather than written by him.

Consider an "iPhone app blog" or a "Indie music blog". These are blogs about subjects, and cannot of course be written by an Iphone app or by some indie music. Similarly, blogs can be about people, but not written by them: for example,

"Top 10 Donald Trump Blogs and Websites to Follow in 2019" - https://blog.feedspot.com/donald_trump_blogs/

None of the listed blogs are written by Donald Trump, but they are referred to as "Donald Trump blogs".

"The blog of Bill Gates" helps to disambiguate this situation.