I suspect that those who say that fun, best and favorite are synonymous are not being completely honest. Fun, greatness, and personal preference are not independent but they are different.
As others have mentioned, "best" may often incorporate how others view the course, while "favorite" does not.
In my opinion, there are certain characteristics that courses in the "best" category should have that are unnecessary for "fun" or "favorite" courses. The "best" courses should be good tests of golf, meaning good shots should be rewarded over marginal ones, and marginal ones over poor ones. On the "best" courses, the best players should differentiate themselves in medal play. There is also some element of length and difficulty involved in great courses over fun or favorite ones. These courses do not have to be long, but they should not be overly susceptible to low scores or easily overpowered by skilled players.
Fun and favorite courses need not meet any criteria. They can be 5000 yards with all sorts of blind shots, funny bounces, funny lies, quirky holes, etc.
My lists of most fun, favorite, and best courses played overlap quite a bit. However, I frequently walk off a course and say that it was fun but not a great course, or vice versa.
Bethpage, Torrey, and Pinehurst are great courses that many don't think are very fun.
Pasatiempo, Ballyneal, Yale, Eastward Ho!, and Rustic Canyon are right at the top of my fun list but no my "best" list (I'm going to GCA hell for saying this.)
There are very few courses that are very strong in both categories: Prairie Dunes, Merion, Pacific Dunes . . .