The path example is extremely fertile ground for this kind of discussion! It is definitely true that both paths encode the information that one's house is connected to the local store. But is that all they encode? Homotopy theory is all about the different paths between two points, and it tells us some quite interesting things! In particular, if you have two paths from point A to point B, you can ask: can you smoothly animate an image of the first path into an image of the second, such that every still frame in-between is also a legitimate path? (If you can't, that tells you that there's some form of hole in between them!)
In the house/store example, a path is also a witness to the fact that, if you perform a road closure anywhere not on the path, then connectivity is preserved. Simply stating that the two points are connected doesn't tell you whether it's safe to close a road! Moreover, taking the two paths together tells you that performing a single road closure that only affects one of the paths will still leave a route you can take.
In both examples, if the paths were logically interchangeable, you wouldn't be able to get more information out of the both of them than you could from just one. But because they aren't equivalent -- because each contains some information that the other does not -- we can deduce more from both together than from either individually.
In the house/store example, a path is also a witness to the fact that, if you perform a road closure anywhere not on the path, then connectivity is preserved. Simply stating that the two points are connected doesn't tell you whether it's safe to close a road! Moreover, taking the two paths together tells you that performing a single road closure that only affects one of the paths will still leave a route you can take.
In both examples, if the paths were logically interchangeable, you wouldn't be able to get more information out of the both of them than you could from just one. But because they aren't equivalent -- because each contains some information that the other does not -- we can deduce more from both together than from either individually.