Both terms can be used as nouns. “Remaining,” as a noun, is an act by which someone or something remains. For instance, in the sentence “The remaining of two natures makes his acts questionable,” “remaining” is the noun. “Remainder,” used as a noun, is the part or parts that are left when something is removed.

Remains and remainder are things. They mean basically the same thing but some are used in certain context. Remains is often used to describe someone who has died or something that's been destroyed. Remainder can just be something that's diminished, like "we are the remainder of the cake from yesterday." You shouldn't use remains there.

As the word suggests, a remainder is something left over, or ‘remaining’, after you’ve completed a division problem. Once you’ve gone through all the steps of dividing, you might still have an extra value remaining. This always happens when one number is not completely – or evenly – divisible by another.