They are pretty much synonyms in chinese but i will say 办法 is used more often in casual situations. like 没办法(=there is no way or there's nothing i can do) and 方法 is more suitable for a detail or technical methods for doing something such as 化学实验方法 (methods for chemical experiments). The reason there are two similar words like this is 办法 is a chinese vernacular word that has always been very very familiar to chinese speakers and you can see that from the fact that 法 is often pronounced in 轻声. 方法 was a borrowing from Japanese and in earlier days they were mainly used in technical documents and translations, so the difference still lingers on. It's a bit like "way" and "method" in English. One is originally English and the other one is from Latin, I believe "way" sounds more familiar to english speakers.

1、办法 means method but relate little with the detail of the methodology, what's more, when saying 办法, we always have strong emotions towards the people doing it. for example: 这是个好办法(praising the people doing it, indicating"you are clever!!") 2、方法 relates with details of the method and often without strong emotions. 训练方法 we say this without clear emotion of approval or disapproval

A: 怎么办？ B: 没办法 = I've no idea. or 我还没有找到正确的办法, Similar to： I am trying, but still not find the right way.

They are synonyms in Chinese .