They are basically the same but they might have slightly different meaning for the people using them.
Sore back could refer to doing an activity a day or two before that left your muscles feeling overworked and tired and so you feel sore but you aren't worried about your back having a problem.
If you say I have a backache it could mean I have done something to harm or hurt my back or something is wrong with my back (where as if your back is sore it could be from a good thing- maybe a difficult or new workout that left your muscles sore or the mountain climbing you did for fun). A backache would be considered worse than a sore back.
I would probably use them in different ways to convey my meaning more clearly, but I also believe a lot of native speakers would use them interchangeably and not think much about the difference in meaning.
Good luck! Hope this helps. Let me know if you have any other questions! :)