How each model works
Fixed price (or lump sum): the contractor prices the whole defined scope up front and commits to it in a written contract. If the labour runs long or the crew hits a snag inside that scope, that's the contractor's problem to absorb - the number you signed doesn't move unless you change the scope.
Hourly / cost-plus (time and materials): you pay for actual hours worked plus materials, often with a percentage markup. The starting estimate is just that - an estimate. The final cost is whatever it adds up to, and you carry the risk of every delay and inefficiency.
Where the money really moves
The difference isn't the hourly rate - it's who owns the uncertainty. On an hourly job, a slow crew, a re-do, or a supplier delay all bill to you. On a fixed-price job, the contractor priced that risk in and now has every incentive to work efficiently, because overruns come out of their margin, not your bank account.
That's also why a fixed price requires real scoping up front. A contractor who fixes a price has to think through the whole job before quoting - which is exactly the diligence you want before demolition starts.
The honest trade-offs
Fixed price isn't automatically cheaper on paper - a contractor prices in a contingency for the risk they're taking on, so a smooth job might have cost slightly less hourly. What you're buying is certainty and a capped downside. Cost-plus can win on very open-ended or discovery-heavy work (a full gut where no one can see what's behind the walls yet), where any honest fixed price would carry a large contingency anyway.
The danger zone is a fixed price built on a vague scope - that's how 'change orders' become a second, unpriced invoice. The protection is a detailed written scope, and a written rule for how any change order is priced and approved before the work happens.
How we do it
We work fixed-price on kitchen and bathroom renovations because it aligns our incentives with yours: we absorb the overrun risk, so we scope carefully and build efficiently. The price on your signed contract is the price on the final invoice. The only thing that changes it is a change you request - priced and approved in writing before any extra work starts. Your completion date is in the same contract.