In the examples you give, there is virtually no difference in meaning between using 'something' and 'anything'. But in general, the word 'something' suggests that you have one specific thing in mind, while 'anything' is much more open.

Anything and something have the same meaning, but anything implies more than one thing. Something implies just one thing. In the first sentence, it would be "he asked if there was anything else that he could do" because, since he was being polite, he would ask if there was more than one more thing that he could do. And in the second sentence, it would be "If there is something wrong, tell me." I'm not quite sure why it is that one in this instance, but it sounds better that way. Does that make any more sense?