This is really just a comment that is too long. Also, I will approach this from the view of knot theory.
Every two knots are homotopic, since they are just different embeddings of $S^1$ in $S^3$. So as a knot theorist, we use ambient isotopy as our equivalence, which is a homotopy where every intermediate map is a homeomorphism. (This is equivalent to the other definitions you might see.) But for either of these two ways of making maps, we can view this happening in S^3 over time.
Now for concordance. We say to knots are concordant in exactly the same way as you define above. But when we think about this in a geometric way, we don't view this as happening in $S^3$ over time anymore. What we do is let the two knots lie in 4 dimensional space $(x,y,z,w)$ where $w\in [0,1]$, one at $w=0$ and the other at $w=1$. And in this space, if we can find a surface that connects the two knots that only intersects the 3 dimensional space at $w=0$ and 1 where the the knots do. For knots, this surface is necessarily a cylinder.
When I first saw this definition, I naively thought this would mean every knot was concordant to every other knot, but this is not true. My erroneous thought was that in 4 dimensions, all knots are trivial, which is what concordance sees. While my thought is true for a knot in 4 dimensions, the difference is that the surface cannot lie in the 3 dim space either knot does. So there are knots which are not concordant to the unknot and knots which are. But all knots are homtopic to the unknot, as I stated above.
I think that PVAL-inactive said it best to actually answer your question in the comments, but I hope I gave some intuition.