Want/need + verb + -ing/ed
The first sentence is incorrect. The second one is correct because it is logically formulated : Maybe they don`t know the difference and [they]need it [be] explained, i.e. They need that the difference be/should be explained to them.
The first sentence is perfectly correct in British English.
Indeed, the first sentence is correct as it stands. The second is incorrect.
Teia, you can't just go adding words to make your favourite sentence correct.
Yes, the second sentence would be better if it had 'to be' inserted: 'Maybe they don't know the difference and need it to be explained', but this isn't a game of 'see how you can improve the sentence', it's a straight choice between the given options.
Teia is right; the first version is incorrect, although many would use this construction informally. The second is correct but not for quite the reason she specifies. It is correct because 'explained' is to be treated in this case as an adjective, the object of which is 'it'. You cannot treat 'explaining' in similar fashion.
I'm a teacher of English as a foreign language and was surprised to find examples like the first sentence in a textbook published by Oxford University Press. As an American, I have never used the pattern in the first sentence, not heard it used before. I would have said it was incorrect in American English, but apparently it is acceptable usage in British English if Oxford University endorses it in an ESL textbook.
The first is perfectly correct and standard in British English, though not in American English.