Is this just a matter of what words we use to describe the situation, or are the two situations really different somehow?
Suppose I and some nearby object are "really moving" away from each other at 1 m/s. Now suppose that we aren't "really moving", but that "the space between us is expanding" in such a way that the distance between us is increasing at 1 m/s. Do I notice anything different? If so, what?
I think there is a real difference, but it's been three years since I took a GR course and I can't say I really understood it much at the time, anyway here goes...
In all theories of mechanics, whether Newtonian or Einsteinian, special or general relativity, we have to have a mathematical description of motion. We do that by labelling the points of space with coordinates. Now that immediately introduces a difference between the
mathematical descriptions of the two cases you've described. In the first, the coordinates of the nearby object (as measured by you) are changing, whereas in the second case they are not. Now here's a

bit: in the second case, even though the coordinates are not changing, the
distance from you
is. This sounds bizarre at first. The thing to realise is, the distance between two points is generally NOT simply the difference in their coordinate. (Let's assume we have just motion along one coordinate axis, say the 'x' one)
This point is actually obvious for, say, spherical polar coordinates on the Earth's surface. If you move one degree of longitude to the left, you can't necessarily say you've moved a metre to the left. It depends what latitude you're at. You could have moved 110km if you were at the equator, but only a few inches if you're near the North Pole.
In general the distance between two points is given by the 'metric', a function of the change in coordinates, and of the coordinates themselves. In abstract geometry this metric can be any godforsaken pathology, but it takes a particular form in general relativity.
So basically, in the first case the object is moving through space, and you can take the metric to be that for simple Euclidean space with a Cartesian set of axes. In the second case, the object is not moving through space, but the metric itself is changing over time.
So that's the MATHEMATICAL difference between the two situations. As for what the PHYSICAL difference is (ie the answer to your question!), I'd have to say... er...

no idea, sorry! I'll keep thinking about it.