They have the same meaning. In equivalent to', 'equivalent' is an adjective. In 'the equivalent of', 'equivalent' is a noun ('the' is necessary). Walking 10km is equivalent to running 4km. A 10km walk is the equivalent of a 4km run. So it is simply the sentence structure which differs.

Good question, but a bit difficult to answer. "equivalent to" is much more common than "equivalent of". We use equivalent to when we name two things and say they are equal or the same. "One Canadian dollar is equivalent to (equal to) 3.93 Brazilian Real." "The number 10 is equivalent to 100 divided by 10." "equivalent of" is used with the word "the" "One Canadian dollar is THE equivalent of 3.93 Brazilian Real." "The number 10 is THE equivalent of 100 divided by 10." - Something is equivalent to something else. - Something is THE equivalent of something else. Hope this helps a little....