difference between anyway, by the way

anyway - (ямартай ч, юутай ч)  - in spite of the fact that you have just mentioned. 
- used when you are ignoring details so that you can talk immediately about the most important thing:
1)in spite of the fact that you have just mentioned:
    --- This idea probably won't work, but let's try it anyway.
2)used when adding a remark which shows that the fact just mentioned is not important:
     ---- They didn't have any trainers in my size, and anyway I'd already 
3)spoken used when you are ignoring details so that you can talk immediately about the most important            thing:
     ---- He got lost and spent hours looking for the station, and anyway it was past midnight by the                 time he got home.
4) spoken used when you are changing the subject of a conversation or returning to a previous subject:
         ---- Anyway, how are you?
by the way - (ингэхэд)used to introduce a comment or question that is not directly related to what you have been talking about

(also by the by/bye) (informal)used to introduce a comment or question that is not directly related to what you have been talking about      ---- By the way, I found that book you were looking for.      ---- What time is it, by the way?

difference between by the way & anyway

• Both "bye the way" and "anyway" are used to introduce different topics into a conversation. This is what is happening in your first 2 examples and either can be used.
  
  "Anyway" is also used to emphasize what is being said. I this is what you need in your last example. As you are not introducing a new subject use "anyway" rather than "by the way". 
  
  Your sentence sounds a little unnatural. I would change it round and say:-
  I threw your shoes away, you won't be needing them any more anyway.
  
  (them as shoes are plural)

• This is a common question, if you google it you will find lots of answers.
  
  Generally, the difference is that "by the way" moves away from the topic and "anyway" reverts back to a previous topic. But "anyway" can also be used to end a discussion about a topic and introduce a whole new topic. "By the way" is never used when you want to go back to an older topic.
  
  Imagine two people talking about the weather. Suddenly one of them says: "Anyway.. where's Dan?". The discussion is about the weather is now over and the new topic is about where Dan is. If he would have said "By the way, where's Dan?" the discussion about the weather could continue after the other person told him where Dan was. It doesn't have to though.

• by the way - moves onto a new topic (that you just remembered you wanted to say)
  
  anyway - moves back to an old topic, that has been interrupted for a short while.
  
  What are you doing, by the way? = Interesting stuff you were talking about, but let's talk about what you're doing.
  
  What are you doing anyway? = We were talking about this before, but you didn't answer, so tell me now.
  
  The two can come very close in certain practical situation.

• So, “anyway” is used when we want to change the subject completely, or when we want to bring back the original subject after being interrupted. For example:
  A: …so that’s why I didn’t like that movie.
  B: Oh I see. I won’t go see it then. Anyway, where do you think we should have dinner tonight?
  The expression “by the way” is used when we suddenly remember something that we want to say to someone. For example:
  I’m going to the supermarket after work today. Oh, by the way, Harry wants to know if you’re coming over for dinner this weekend