What's the difference between "what you like the most" and "what you enjoy the most"?

I can't think of any sentences where the choice of 'like' or 'enjoy' in the above phrase would make any difference to the meaning - so basically no, there's no difference.
In general, 'enjoy' has more limited usage than 'like' and is normally used for activities, so :
I like football or I enjoy football
I like going to the cinema or I enjoy going to the cinema
- in both these cases 'enjoy' puts a little more emphasis on being engaged in an activity than 'like', but it's only a nuance ;
but
I like wild flowers - much more likely than 'I enjoy wild flowers' (though the latter is not impossible)
But in the examples in the OP, the nature of the thing or activity that is appreciated should already be apparent from the rest of the sentence, so the choice of verb after the 'if' makes no real difference that I can see.

Woops, typed too fast in #2, I should have said
'the choice of verb after the 'WHAT' makes no real difference that I can see' - not after 'if' as I put.
#3 yes there is a difference - ' I enjoy HB' has a nuance of appreciating some kind of activity associated with Halle Berry, ie. something more than just liking the person herself - so from the context it probably suggests you enjoy watching her films ( for example ).
But I still can't see any real difference in the examples given in the OP.
I would say that you can like food items to which you are allergic, so you cannot say that you enjoy them (if they make you puke).
I have a friend who loves curry but who can't eat it anymore because she has an allergic reaction to it.
And, harking back to an old discussion (possibly on another branch): "Customers like discounts" is not the same as "Customers enjoy discounts."

Not TOO deeply delving into this, but, are you suggesting that it is possible to LIKE something without enjoying it?, or that it is unlikely that you would enjoy something without necessarily LIKING it?

The discussion that shilgia is talking about (on Get Stuffed maybe?) involved the use of "enjoy" to mean something like "benefit from" or "have available for (one's) use." "Our employees enjoy generous vacations" doesn't mean that the company polled the employees to see if they all had a good a time. There's a quote in my dictionary "At one time the white elephant enjoyed immense symbolic significance in the East." That doesn't imply that the white elephant liked it.
I think what may have got the discussion going was my complaining about shop signs saying "Please enjoy your food and beverages outside" rather than "No food or drink." I said that it isn't possible to ask someone politely to enjoy something. You could ask them to pretend to enjoy it, but not to enjoy it. Shilgia then brought up an examle involving this other meaning of enjoy and it gave me some trouble. I'm still mulling it over.
(Actually I've forgotten what her troublesome example was, but don't tell her that. It may come to me.)
Edited by: VinnyD