The physical differences are many. The most obvious is that they are 2 different objects.
Let's assume they have identical mass distribution because they are composed of identical particles. Their location and orientation still differ.
Regardless of relativistic effects, one is described as still, the other in motion. Without specifying a single observer frame, I assume a majority of local observers would reach consensus on these labels. That makes it valid to consider those observers as data points in themselves. The gravity of the still asteroid will have affected the motion of local observers in a uniform way. It has a "sphere of influence".
The gravity of the moving asteroid will have affected the motion of a different set of local observers in a non-uniform way. It has an "egg of influence". The faster the asteroid is moving, the longer and thinner the "egg" becomes. The wake of the egg vs. the wake of the sphere will be discernible in a snapshot. Rather than just looking at reality through the frame of one observer, we need to consider the perspectives of all "eyewitness" observers.
Like a detective canvasing a crime scene to decipher what has happened; 2 observers may have wildly differing stories about what happened due to their perspective, equally valid, equally true, but there is a larger truth. The ground truth that satisfies all observations.

Captain Obvious wonders if this may be a more intuitive way to explain spacetime, time dilation, foreshortening. An object approaching the speed of light would have such a thin "egg" that it does not affect objects perpendicular to its path at all.

The physical differences are many. The most obvious is that they are 2 different objects.
Let's assume they have identical mass distribution because they are composed of identical particles. Their location and orientation still differ.
Regardless of relativistic effects, one is described as still, the other in motion. Without specifying a single observer frame, I assume a majority of local observers would reach consensus on these labels. That makes it valid to consider those observers as data points in themselves. The gravity of the still asteroid will have affected the motion of local observers in a uniform way. It has a "sphere of influence".
The gravity of the moving asteroid will have affected the motion of a different set of local observers in a non-uniform way. It has an "egg of influence". The faster the asteroid is moving, the longer and thinner the "egg" becomes. The wake of the egg vs. the wake of the sphere will be discernible in a snapshot.

Rather than just looking at reality through the frame of one observer, we need to consider the perspectives and relative movements of all "eyewitness" observers, and just as important, observers nearby that "didn't see nuthin", and the pattern of the border between them.
Like a detective canvasing a crime scene to decipher what has happened; 2 observers may have wildly differing stories about what happened due to their perspective, equally valid, equally true, but there is a larger truth. The ground truth that satisfies all observations.

Captain Obvious wonders if this may be part of a more intuitive way to explain spacetime, time dilation, foreshortening. An object approaching the speed of light would have such a thin "egg" that it does not affect objects perpendicular to its path at all.