Both "I hope you'll read lots of good books this year" and "I hope you read lots of good books this year" are correct, right?
Is one of them more common than the other? Is there any difference between UK/US English?
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While both phrases are grammatically correct, I think there is a slight semantic difference: using will (in this case, you'll instead of you) puts more emphasis on someone's determination to do something.
To use your example, "I hope you read lots of good books this year" basically means "I hope it works out for you to read lots of good books this year." You're emphasizing the outcome more than the action.
On the other hand, "I hope you'll read lots of good books this year" is a little more forceful; it could be interpreted as "I hope you make the effort to read lots of good books this year." You're emphasizing the action, not just the outcome.
I basically agree with other answers, but have my two cents' worth. "I hope it rains tomorrow" is definitely more natural than "I hope it will rain tonorrow", and, "I hope you will forgive me" is definitely more elegant and effective than "I hope you forgive me". Why? Because "will" keeps its sense of "want", "be willing" in many contexts. "I will answer the phone" is a friendly offer, meaning "I am willing to answer the phone", whereas other futures, like "I am going to answer the phone", sound like "I have decided to answer the phone, so that is final; sit down, sucker!" Or (annoyed) "you will keep leaving your socks on the floor!" "Will" is cognate with German wollen. So in cases where you hope someone is willing to do something, like forgive you, or come to your party, "will" is a very nice little improvement on the verbal base. In cases where willingness can play no role, such as the weather tomorrow, "will" is so unidiomatic a choice that non-native speakers best regard it as wrong.