Both have comparable definitions of "method", "manner", "means", etc. and describe or define a way of doing something. The phrase 没(有)办法 is often used to deflect responsibility by claiming powerlessness. Aside from being more common in colloquial speech, does this make 办法 more general? Anecdotally, 方法 is encountered frequently in instructions. Does this make 方法 more technical?
办法 is simply 办事的方法.
This has nothing to do with colloquialism or technicality. Both are common words used in daily speech and writing.
方法 = method, way or solution
办法 = method for handling or doing something
From the above, it is not hard to see that, 方法 is more generic than 办法. 办法 can only be used on methods that are actionable. 没办法 implies there is no way something can be done. 没方法 means there is no solution yet.
If the method is non-actionable, you would have to use 方法. For example:
What are the common ways for a knight to defeat a bishop at the end-game?
I don't think you guys are completely understanding it, there is a philosophical difference in the two words. It's possible to 有方法但是没办法. 方法 refers to the general method or way something can be done. 方 means prescribed, 法 here can be understood as way.
So 方法 is the general or prescribed way something is done that humanity has discovered. 办 means doing，法 here also means way. 办法 refers to someone specific actually doing it. For example:
我的病其实治疗方法很简单，就是每天跑步。但是没办法，因为城市太拥挤所以没有地方跑。Here, I know the way to cure my illness, but I can't specifically carry it out due to individual circumstances.
方法 - Usually we used to express it by giving/request "method" to solve the problems.
Example : 你用什么方法去解决? what method do you use to solve it? Example : 我觉得这方法不错? i think this is the good method.
办法 - Usually we used to express it by giving/request "idea"
Example : 你有什么办法来解决这件事吗？ Do you have any idea how to solve this problem? Example : 我有办法！I have an idea!
To translate simply, 方法 means "method" / "manner" / "means" (as you said), and 办法 means "solution".
Note, however, that 办法 only applies to solutions that involve a plan of action. You cannot, for example, call x = 3 a 办法 to the equation x + 7 = 10. Rather, that would be a 答案 ("answer").
Translate this way for every context, it'll simply make sense.
- 办法 is more about solving problems (solutions).
- 方法 is more about the ways (methods) to do something.