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.

Trust the dictionary. They express different moods. by the way = 对了,.... 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.

ANYWAY CORE-MEANING: an adverb meaning no matter what the situation is Anyway, we have to pay whether it was accidental or not. Recycling, according to some anyway, is the best way of teaching respect for the environment. adv 1. in any case: no matter what Don't worry about the damage; I was going to buy a new one anyway. 2. regardless of something: in spite of the situation I knew it would be a sad movie but I went anyway. 3. in careless way: in a careless, haphazard, or lazy way According to my mother, packing is a skilled operation, not throwing your clothes into a case just anyway. 4. an·y wayby any means: in any manner or way (informal) We have to teach our children moral values anyway we can. BY THE WAY adv 1. very much: to a considerable degree or at a considerable distance (informal) way out of our price range 2. △very: extremely (slang) That's way cool! NOUN 1. manner or method: a means, manner, or method of doing something I'll do it my way. 2. respect: a feature, aspect, or example of something In some ways, they're very similar. 3. path: a path or physical means of getting from one place to another The way out is through here. 4. door or opening: a door or opening leading or providing access to or from somewhere came in the front way 5. journey or route: a journey or the route followed or to be followed on my way home 6. progress through life: progress or a path through life and its experiences or difficulties 7. Waystreet: a street (often used in place names) 8. direction: a direction, e.g. left, right, up, or down 9. manner of placing: the manner in which something is placed, packed, or arranged, or the direction it faces HOPE THAT ANSWERS YOUR QUESTION

You should trust your dicts, that is, their Chinese meanings in that case will confuse you as well like the 2 English phrases do.

Jon is essentially correct, but it's not always true in that way. I think it depends on the speaker - some use it exactly the way he said, but some (like myself) use them interchangeably. These two phrases aren't really set in stone, because they're not truely proper English. However, they're not slang either... they are somewhere in between.