I think it would be interesting to write both of them in a way that only by switching some lines of code would give you one algorithm or the other, so that you will see that your dillema is not so strong as it seems to be at first.

I personally like the interpretation of BFS as flooding a landscape: the low altitude areas will be flooded first, and only then the high altitude areas would follow. If you imagine the landscape altitudes as isolines as we see in geography books, its easy to see that BFS fills all area under the same isoline at the same time, just as this would be with physics. Thus, interpreting altitudes as distance or scaled cost gives a pretty intuitive idea of the algorithm.

With this in mind, you can easily adapt the idea behind breadth first search to find the minimum spanning tree easily, shortest path, and also many other minimization algorithms.

I didnt see any intuitive interpretation of DFS yet (only the standard one about the maze, but it isnt as powerful as the BFS one and flooding), so for me it seems that BFS seems to correlate better with physical phenomena as described above, while DFS correlates better with choices dillema on rational systems (ie people or computers deciding which move to make on a chess game or going out of a maze).

So, for me the difference between lies on which natural phenomenon best matches their propagation model (transversing) in real life.

4I cannot understand why this answer has 27 upvotes and it is exactly the merging of 2 other answers, which by the way are simply general thoughts about... – nbro – 2015-05-31T15:06:33.863