Both expressions describe a flag flown below its usual position as a sign of mourning or respect. Half-staff is the wording used in the U.S. Flag Code. Half-mast is traditionally associated with a ship's mast and maritime settings, although people also use it in everyday speech about flags on land.
The Wording Is Not the Instruction
A search for either expression may lead to the same notice. What matters for your display is the actual authority, affected flag, geographic scope, covered facilities, and dates. A state-flag instruction does not automatically establish the status of the U.S. flag. A local facility's display does not establish a nationwide order.
Where Half-Staff Is on the Pole
The Flag Code describes half-staff as halfway between the top and bottom of the staff. It also describes first raising the flag to the peak briefly, then lowering it to half-staff, and raising it to the peak again before lowering it for the day. Check the specific order and practical display conditions if you have multiple flags or a setup that cannot be lowered.
Find Today's Answer
Use the national status page and your state page to review applicable instructions and official sources. We use half-staff in our main labels while addressing half-mast questions in the same guides, so you do not need to guess which phrase to search.
Source: U.S. Flag Code, 4 U.S.C. 7(m). Related: Lowering and Raising a Flag; Displaying Multiple Flags.