These nouns have similar meanings, but a careful examination of those meanings reveals how attention to precision of language can improve the professional image of your writing. Balance has two meanings. The first is a state of equilibrium. The second is specific to accounting. In that context, balance means the amount left in an account after accounting for all deposits and withdrawals. Remainder has a similar meaning but applied to any context; it the amount left over, whether you are talking about a financial account or anything else.
Consider the example sentence in the cropped graphic above. Because balance more precisely references the context of a financial account, remainder is more appropriate when discussing alloy composition.