Which expressions should I choose to say “thanks for your help”?
In my opinion, 帮忙 and 帮助 are different.


  帮忙: [verb]to help some do something .This word itself has declared its object[忙] already. I think the phrase "do sb a favor" is a more appropriate equivalent in English. 
  
  帮助: [verb]to help. [noun]help. This word needs an object


You would say, 


  你可以来帮忙吗？ Could you come and help? 
  
  你可以帮助*我*一下吗？Could you help me?
  
  我帮不上忙。 I am not able to help.
  
  我无法帮助*你*。(Actually, 我帮不了你 is more common) I am not able to help.


Imagine that you are going to prepare a meal in your house,to entertain your friends, but you have a lot of work(buying, washing, cutting, cooking...) to do
, so you ask me for help. You would say,


  Huang,明早你可以来帮忙吗？Could you please come and help(do me a favor) tomorrow morning?  By using 帮忙, you should not express its object, and that doesn't matter, because in the context, I understand what I can help you. I say,
  
  没问题。我能帮助你什么呢？=没问题，我能帮什么忙呢？No problem! What can I help you?
  
  你可以帮助我去超市买菜。 You can help me buy the foods in the supermarket. Here, you won't say 帮忙, because you have to express the object explicitly.


Now, for your question, I think the 2nd sentence is more common, though the first one could be understood. I guess you would not say "Thanks for your doing me a favor" in English, would you? 

You can also say other words to express this meaning,


  谢谢(or 谢了 if you are familiar with the interlocutor)
  辛苦你了
  辛苦了
  麻烦你了
  麻烦了