How to think about concordance? Two continuous maps $f, g: A \to B$ are homotopic if there exists a continuous map $H:A \times [0,1] \to B$ such that $H(x,0)$ = $f(x)$ and $H(x,1) = g(x)$. I understand that one should think of $H$ as a continuous deformation of the map $f$ to the map $g$, and there are gifs on the wikipedia page and all is right with the world. 
Two continuous maps $f, g: A\to B$ are called concordant if there exists a continuous map an embedding $J: A\times[0,1] \to B\times[0,1]$ such that $J(x,0)$ = $f(x)\times\{0\}$ and $J(x,1) = g(x)\times\{1\}$. Is there an intuitive way to understand concordance? Or are there some simple, illuminating examples of maps that are concordant but not homotopic? 
(edit: corrected the definition of concordance.)
 A: 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.
