Some fields don't care which way you went, only where you ended up. Most do care — and one famous one only pretends not to.
Drag either endpoint. On a gradient field the two totals stay locked together no matter where you put them.
Drag either endpoint and compare the two running totals.
Two routes, same start, same finish. Work out the work done along each. For most fields the two answers differ — which route you took mattered.
For some fields they don't, ever. Those are the gradient fields : fields that are the gradient of some scalar potential, whose contours are drawn faintly behind. For those, the integral collapses to φ(end) − φ(start) — just a height difference, blind to the route, exactly like the fundamental theorem of calculus one dimension up. Gravity is the famous example: carrying a box up the stairs or up a ramp costs the same.
There's a handy test for whether a field is one of these. Bending a path slightly sweeps out a little region, and Green's theorem says the change in the integral is the curl inside it. So no curl anywhere means no change, ever — curl F = 0 is necessary.
And now the awkward one. That last field has curl exactly zero everywhere it is defined, and yet the two paths disagree — because it isn't defined at the origin, and the region between the paths straddles that hole. Green's theorem never applied. This is why the curl test comes with the words simply connected attached: the shape of the domain is part of the theorem, not fine print. Drag both endpoints to the same side of the origin and the disagreement vanishes.