割引 and 値引き: a conditional discount and a reduced selling price
Compare a student discount with a reduction for scratched goods, without dividing the words solely into percentages and fixed amounts.
A discount linked to conditions or a rate
割引, read わりびき, means a reduction from the original amount. この店には学生割引があります。 says that the shop offers a student discount.
割引の対象は学生だけです。 says that only students are eligible. Read the named discount together with its actual conditions.
Lowering a selling price
値引き, read ねびき, means lowering an established selling price. 小さな傷があるので、店が値引きしました。 says that the shop reduced the price because of a small scratch.
値引き後の価格を確認しました。 says that the price after the reduction was checked. Distinguish 前 and 後 when comparing it with the original price.
A shared meaning and overlapping labels
The words can overlap in contexts where prices are reduced. Do not make an absolute rule that 割引 always uses a percentage and 値引き always a fixed amount.
値引きをお願いしましたが、店員は断りました。 says that a price reduction was requested, but the shop assistant refused. Separate making a request from obtaining a reduction.
Does 値引きをお願いしましたが、店員は断りました。 establish that the price went down?
Show the answer and why
No. A reduction was requested and refused; no completed reduction is stated.