The two terms get used interchangeably, and that costs contractors money when they order the wrong deliverable.
A quantity takeoff answers one question: how much of each thing is there? Cubic yards of concrete, squares of roofing, linear feet of pipe, counts of doors. It carries no dollars. It is what you hand to a supplier for a material quote, or to your own team to price with your rates.
A cost estimate takes those quantities and applies material costs, labor productivity rates, equipment costs, subcontractor pricing, overhead and profit. It answers a different question: what will this cost me to build?
Which one should you order?
If you have your own cost database and crews you have run for years, order the takeoff and price it yourself – your rates are better than any published average. If you are bidding a trade or a region you do not know well, or you need a budget number for an owner, order the priced estimate.
Most contractors end up using both: takeoffs for the trades they self-perform, and full priced estimates for scopes they are subbing out or bidding cold.