in fact,the difference is not great because they are close means together

In my mind there is no or very little difference in saying "walk across" or walk over" the bridge. I would happily use both.

They are so similar you could use either and you would not be corrected or challenged. If anything, I would use "walk across" if I was giving directions as it sounds more like you are on a path to somewhere, whereas if I was taking a leisurely stroll with no immediate destination I would say "walk over" as it is a little more gentle and less directive, but that is more a personal preference than a rule.