le mieux vs le meilleur

It seems to me that both mean better and best. Please explain and give an example when to use le mieux and le meilleur. 

Oh, not at all. It might help to think of it as a chain of either bon or bien, so you know which one to use:

bon->meilleur->le meilleur (good, better, best)
bien->mieux->le mieux (well, better, best)

Bon is pretty much used as an adjective, bien as an adverb.

Les cerises sont bonnes, mais les raisins sont meilleurs. Mais, non, les fraises sont les meilleures.

Note that because bon is an adjective, all its forms will agree with number and gender of the noun it's describing or modifying.

Jacques parle bien français, mais Thomas parle mieux français. Mais non, je trouve que Sandrine parle le mieux.

Because bien is an adverb (describing/modifying parle in this case), there's no agreement necessary, bien/mieux/le mieux are invariable.

Merci mille fois, torusan,

Non one couldn't have explained any better, with  great examples no less! I don't recall RF explained it and that's why I got confused but then it doesn't take much. 

 

Haha, yeah I think this is one of those "think in French" things too, because in English, we use the same "superlatives", so we don't necessarily think about how it's being used. :)

I'm too lost and confused to entertain "think in French". I have more pressing issue to deal with such as the superlatives. 

You lost me a bit; we don't use the same superlatives. For some nouns we use the suffix -er and -est like pretty but for others we use more and most like beautiful; two ways for beginning English learners. 

French, as you pointed out and gave great examples only the adverbs are like English but with adjectives it's a different ball game. We have to think about the number and gender. I still make beginner mistakes when I take the tests even though I knew the answers, forgetting to dotting the i's and crossing the t's so to speak. 

Circling back to my first statement may be we should all learn to think in French on day one if that is at all possible! :-)

Ah, I was thinking that we also say things like "I play the piano well, but he plays better." In that case, we're modifying the action of "playing", so bien->mieux.

But when we say "The chocolate cake is good, but the apple tart is better", we're qualifying the type of dessert, not the action, so it's bon->meilleur.

 

I'm having a "temper tantrum" with 6.8. This is a good break for me!

All the numbers were correct but in red numerals, NOT written out as the lesson whether they were hours or amount of money. For example, Monsieur Riviere asked combien coute ce vol s'il vous plait and the airline person said "ce billet est a mille cinq cents euros Monsieur. (Sorry about all the missing accents!)  Mine came back correct except 1,500 € (and flight hours) in red. I don't know what needs to be changed to get all of them in green. I listened to my recording and of course I thought they sounded the same. The nuance that a beginner couldn't hear, of course! 

By the time I get the bon-meilleur and bien-mieux roll off the tongue in seconds I should get a BA in this area, :-)

Merci mille fois! 
 

Hey ML - that number thing is a known bug. :(
Don't worry about it - if it sounds good enough it probably is.
They're working on fixing it, don't put loads of time in ... :)