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.

By the way introduces something alongside (by) your path (the way). . anyway (which seems to be anyways to Americans) adds something relevant to your path, regardless of something, anyway we must .... = so regardless of any choice , we must ...

In my opinion, "Anyway" is refer to "by all means." "By the way" is much likely to "On the other way" or "In addition."

Perfect, thank you so much! ☺️