Is there any difference between "Is there something wrong?" and "Is there anything wrong?"? Also, you would say "He would like something to drink" but "Would you like anything to drink?", right? I'd appreciate if any one could explain the usage of "something" and "anything".
To me, we typically expect a positive response when using "something".
It is common for "something" to be used in the affirmative and sometimes when asking a question with the expectation of a positive response. This is not a firm rule, just be aware of the context.
She bought something at the flea market. Would you give him something to do?
It is common for "anything" to be used in the negative or interrogative and more commonly than"something" with questions.
He didn't want anything to do with those girls. Is there anything that I can get you? (You can also say, "Is there something that I can get you," but using "anything" sounds a bit more polite.)
Here's the source.
In your examples, "Is there something wrong" probably implies you feel something might be wrong and want to confirm with others while "Is there anything wrong" is just a general inquiry whether anything is wrong at all. "Would you like something to drink" is likely to be heard in a bar where you are expected to order something to drink while "Would you like anything to drink" might just mean you want to know if a person is thirsty or not.
Furthermore, something indicates finiteness while anything doesn't place any restriction on the scope or quantity. That said, "Would you like something to drink?" might be synonymous with "Would you like to choose one from our (finite)collection of drinks on the shelf" while "Anything to drink" might mean anything drinkable that you could imagine.
"Would you like anything to drink" also sounds more polite to me as it doesn't potentially impose a positive response on the person(e.g., customer).
As others have said, there's no difference in meaning between anything and something, but there are important grammatical differences.
(1) Only something is used in independent positive statements. We can say Something is wrong, but we can't say *Anything is wrong.
(2) Only anything is used in negative statements. We can say I don’t know anything about it, but we can’t say *I don’t know something about it.
(3) Both can normally be used in questions, but there are exceptions and nuances. If you return from the store, I would ask you Did you buy anything? rather than Did you buy something? A speaker who didn’t hear what another person was saying would ask Did you say something? not Did you say anything? You might ask that in reply to someone who felt offended by the action or words of a third person.
(4) Something and anything can be used in making offers, but there are differences. If you were offering a wide choice of something, you’d say Help yourself to anything you want and not *Help yourself to something you want. And Would you like something for your birthday? is more forthcoming than Would you like anything for your birthday?
The ‘Longman Student Grammar of Spoken and Written English’ (LSGSWE) calls pairs like these ‘assertive’ (something) and ‘non-assertive’ (anything). It confirms that non-assertive forms are associated with negation, including implicit negation: I rarely find anything to laugh at in his jokes rather than *I rarely find something to laugh at in his jokes. It also points out that non-assertive forms are associated with conditional clauses, temporal clauses introduced by before and comparative and degree constructions. That’s not always the case with something and anything, but it is with the last: I like treacle tart more than anything and not *I like treacle tart more than something.
(The LSGSWE is the stripped-down version of the authoritative 'Longman Grammar of Spoken and Written English'.)
In the context of Would you like..., something and anything are effectively synonyms. I don't think there's any case to be made for any (some) subtle difference in meaning, but in practice it seems something is significantly more common.
But curiously, it's always I would like something to eat when presented as a statement. Almost all occurrences of would like anything to eat here turn out to be things like He asked if we would like anything to eat (reflecting the word as used in the original question).
In short, use either when asking a question, but note that when making a statment we simply don't say I would like anything to eat.
Moving to the more general case, I looked at I want something/anything to..., where it seems clear that something is normally used for straightforward expression of a desire, but anything mainly occurs in "negating" constructions, such as It's not that I want anything to happen.
In your examples anything means the same as something. However the two words may not mean the same in other contexts. For example,
I will give you something
means I will give you a thing
I will give you anything
means I will give you whatever thing you ask for.
In deciding between the two that are equivalent, you can look at the suggestive connotative meaning of the root words some and any. Some is often suggestive of a part of a whole, while any is one or more things out of many things.
Would you like some cake? Did you take any pictures?
If the answers are suggestive of a part:
We have lots of food. Would you like something? (some of the food, not all)
Or if it's multiple things:
We have replicas of the Eiffel Tower, the Statue of Liberty, and the Great Wall of China. Would you like anything? (one or more of those things).
As you can see, for either of those examples using the opposite wouldn't quite make sense.