The difference is that in the fruits example the sailors are not eating all kinds of available fruit, but in the snow storm example all planes in NY are grounded.
Coconuts and the other fruits formed the basis of the sailors' diet while they were shipwrecked on a desert island.
This version is simply incorrect, because you're not specifying what other fruits. So let's go ahead and correct that:
Coconuts and the other fruits on the island formed the basis of the sailors' diet while they were shipwrecked on a desert island.
Okay, so now we have a grammatically correct sentence. But it doesn't mean what the author likely intends, logically. It means that the sailors ate coconuts, and also ate every other type of fruit available on the island. The other fruits on the island says that of the set of all fruits on the island, coconuts are in one group and all the rest are in another, and the sailors ate them all. This just isn't what the author intends to say. The intent is likely that the sailors ate coconuts and a subset of the other fruits on the island. They ate some of the other fruits, but not all. So we must not use the:
Coconuts and other fruits formed the basis of the sailors' diet while they were shipwrecked on a desert island.
It's implied that both coconuts and the other fruits eaten exist on the island, and that there are other fruits that exist as well. And that makes logical sense.
Now, for your planes example.
If you fly to New York, and there is a snow storm and you can't leave the city because the planes are not flying, you're stranded in New York...
Here the author does mean that all the planes in New York are grounded. There's a snowstorm; as far as I know, we haven't invented a commercial airliner that's safe to fly in snowstorms. So if there's a snowstorm and some planes are grounded, all planes are grounded. So here we use the because we are referring to all the planes in the NY airport. (Just like in the fruit example, where the other fruits referred to *all the types of fruit available on the island.)
If you fly to New York, and there is a snow storm and you can't leave the city because planes are not flying, you're stranded in New York...
If you do not use the, then you're only saying that some of the planes are not flying. Of all the planes in the NY airport, only a subset of them are not flying. This doesn't logically follow, as discussed above. So we cannot omit the here. It changes the meaning.
These (terribly drawn) images might help: