I would say the meaning of "the area between two curves"
in a context such as the one in the question
is literally that you have two functions, each of which has been
plotted as a curve in a Cartesian plane with the horizontal axis appropriately labeled so that it is the input variable of both functions;
if that variable is $t,$ then
the area between the two curves from $a$ to $b$
means you draw vertical lines $t=a$ and $t=b,$
and now the two curves and the two lines enclose a region in the plane,
and we measure the area of this region.
But people also use the phrase "area between the curves"
as a kind of verbal metaphor and/or graphical shorthand for a definite integral of the difference of two functions.
The basis of this metaphor is that in many cases we can compute the
area bounded by the graphs of two functions and by two vertical lines
by taking a definite integral of the difference of those two functions.
The problem with the metaphor is that if we integrate $f(t) - g(t)$
over an interval on which $f(t)$ is sometimes less than $g(t),$
we end up with regions of the plane whose area has to be subtracted
from the total "area" in order to have the "area" come out equal
to the integral.
And finally there is the application of "the area between two curves,"
which often comes down to knowing that there is one process of some sort that is increasing a quantity at a certain rate and another process that is decreasing the same quantity at a certain rate,
and the "area" (actually the integral of the difference of those two rates integrated over a certain period of time) is the accumulated effect of those two processes.
So if you have two objects moving along the same straight line,
starting from the same point at time $a,$ and
the velocity of each object is is how fast it is moving to the right,
then if the velocity of object number $1$ is given by $v_1(t)$
and the velocity of object number $2$ is given by $v_2(t),$
$$\int_a^b (v_1(t) - v_2(t))\, dt$$
tells me how much farther to the right object number $1$ will be
compared to object $2$ at time $b.$
This also happens to be numerically equal to the area between the graphs of the two velocity functions from $a$ to $b,$
with the caveats that this works only if we chose all our units appropriately
and that if $v_1$ ever dips below $v_2$ on that graph then
we need to treat parts of the area between the two curves
as "negative area."
In an application like that, my viewpoint is that what people really want to do is to integrate the difference of two functions (which of course is itself a function), and the phrase "area between the curves" is just their way of saying that they would like to visualize this integral by plotting the graphs of those two functions on a Cartesian plane.