Both words can describe people or things which are left behind after the rest has been taken away, used up or destroyed.
Ex: "The remainder of the food will be given to charity."
Ex: "The remaining soldiers still stood strong on the battlefield."
In mathematics, "remainder" can have one of two meanings. When a number is divided unevenly, the "remainder" is the number left over. It is also used to refer to the resulting number after subtraction, often called the "difference."
Ex: "32 divided by 5 leaves a remainder of 2, since 5 only evenly divides into 30."
Ex: "The remainder of 10 minus 7 is 3."
In law, "remainder" is used to refer to an estate that passes into another's possession after the termination of an overarching estate.
Ex: "After his business partner passed away, the ownership of the building passed to him as the remainder."
In publishing, "remainder" is used to refer to books that the publisher is left with after sales have decreased, often sold at a discount.
Ex: "The publisher was pleased to have fewer remainders to deal with this month."
As with the verb "to remain," "remaining" can refer to more than just that which is left over; it can also mean not making progress or continuing unchanged.
Ex: "Remaining undecided on the matter, he gave no opinion either way."
Because "remainder" can be used in a more technical sense, its connotation is more impersonal, so in general, avoid using it to refer to people or animals. In these cases, it is better to use "remaining" or the verb "to remain."