Quick answer

Set-off reduces a claim by a separate cross-claim, a liquidated debt or an equitable set-off so closely connected it would be manifestly unjust to ignore, while abatement reduces the claim itself by the amount the defective work has diminished the value of what was done; the two remain distinct in substance, but under the current section 111 a pay-less notice is needed whenever the payer intends to pay less than the notified sum, so the old idea that a notice was required to set off but not to abate no longer decides the question

Set-off reduces a claim by a separate cross-claim, while abatement reduces the claim itself because the work done is worth less than claimed. A set-off requires a counterclaim that is a liquidated debt, which may serve even if disputed, or, if unliquidated, an equitable set-off so closely connected to the claim that it would be manifestly unjust to ignore it, as in Hanak v Green; and clear words are needed to exclude the common-law right of set-off against a certified sum, as the House of Lords held in Gilbert Ash (Northern) Ltd v Modern Engineering (Bristol) Ltd. Abatement is a common-law defence to a claim for the price of defective work: in Multiplex Constructions (UK) Ltd v Cleveland Bridge UK Ltd the measure was the diminution in the value of what the contractor produced, with the cost of remedial works a factor rather than the measure, and delay or damage to other things could not feature. The distinction once decided whether a statutory notice was needed: in KNS Industrial Services (Birmingham) Ltd v Sindall Ltd a withholding notice was required to set off but not to abate, on the reasoning that one cannot withhold what is not due. Under the current section 111 that no longer holds; a pay-less notice is needed whenever the payer intends to pay less than the notified sum, whichever label applies, so the safe course is to give one for an abatement as well as a set-off.