Two lines on your estimate, two dates on your calendar
The place most people run into this distinction is on paper. Your estimate has a line for footers and a line for foundation walls, and your schedule has two pour dates with an inspection somewhere in between. That is not padding. They are two pieces of concrete doing two different jobs, and the second one cannot go in until the first one has set.
The order never changes. The trench gets dug, and the footer goes in flat across the bottom of it with steel tied in a grid. Before that concrete stiffens, either a keyway gets pressed into the top of it or dowels get stood up out of it. Once it has taken its own weight, forms go up on top, the vertical steel gets tied to those dowels, and the wall gets poured. Two pours, two days, one connected piece of structure.
If your estimate only shows one line, ask rather than assume. It usually means the job is monolithic, with the thickened footing edge and the slab going in together as a single pour, which is a real and common way to build. It is a different answer, not a missing one.
One spreads the load, the other carries it and holds back dirt
Think about what each piece is up against. The footer is up against the dirt below it. Everything the building weighs arrives at the bottom of the wall in a narrow line, and soil cannot take weight concentrated like that without moving. The footer is wider than the wall on purpose, so the same load reaches the ground spread over more of it.
The wall is up against two things at once. It carries that load straight down, which concrete is very good at, and it holds back the ground pressing sideways on it, which is a bending problem and is what the steel inside it is really there for. So a wall is not simply a taller footer. Different job, different reinforcing, and the drawings say so.
A stem wall on a crawl space build is the same arrangement at a shorter height. Wall on a footer, carrying the sill plate and the floor system above it, holding the perimeter of the building at the elevation the plan wants it.
Where the two words get used loosely, and when that costs you
In everyday talk, foundation is an umbrella word for the whole below grade package, footer included, and nobody is wrong for using it that way. It starts to matter when somebody is quoting you, inspecting you, or telling you what has already been done on a site you have not walked.
So when a builder says the foundation is in, it is worth asking which parts. A poured footer with dowels standing out of it and no wall yet is a very different site from a stripped wall waiting on backfill. On a pole building there may never be a continuous wall at all, only individual footers under the posts. On a monolithic pour there is no separate wall either, because the thickened edge is doing both jobs at once.
The reason to be precise is that the inspection, the timing and the next trade all hang off which piece you mean. The footing inspection happens before concrete goes in the trench, not after. Backfill against a wall waits on what is bracing that wall top and bottom. Getting the words straight is the cheapest way to keep those two from sliding past each other on a Friday afternoon.