Hmmm, I thought about this for a few minutes, but honestly I can't pinpoint what the difference is. They are both correct, and seem to be interchangeable. "가를" is not really a word, it's a particle that indicates the function of the preceding word or clause. Here, "가를" and "지(를)" have the same function. It's a kind of a topic marker.

감사합니다!

Both of them are not quite different. '어떻게 생각하시는가' is technically one sentence. That means that 어떻게 생각하시는가 is a clause and used as a noun in sentence. ------------------- There's another situation why it is used like that. [situation: at the mall] A: how about this skirt? You: (didn't listen) sorry? A: I asked what do you think. (어떻게 생각하는지 물어봤어/ 어떻게 생각하는지 물어봤습니다/ 어떻게 생각하시는지 물어봤습니다) As you see, if you didn't listen clearly and then asked once again what someone said, someone can say '어떻게 생각하시는가 물었습니다. Actually there's various expresstion and all of them that I wrote three sentences are same but only different thing is whether does it use polite way. If you confuse, just remember first answer!