바꾸다: it eliminate the something originally it was and then it change or replace it new something. 원래 있었던 것을 없애고, 새로운것으로 바꾸거나 대신하다. 변경하다 : just it change the something differently and then it became new something. 다르게 바꾸고,새로운 것으로 고치다.

As far as I can tell both words have the same meaning (to change) but 바꾸다 is a pure korean word and 변경(變更)하다 is a sino korean word. you can read more about sino korean words here: https://en.m.wikipedia.org/wiki/Sino-Korean_vocabulary