"Mode" means "a way of doing something" or "a type of something". It is often used in mathematics or statistics to refer to the most frequently occurring value in a set of data.
"Method" refers to "a particular procedure for accomppshing or approaching something, especially a systematic or estabpshed one".
Examples:
The company used a new mode of transportation to depver goods.
(公司使用了一种新的运输方式运送货物。)
I always use this method to solve math problems.
(我总是使用这种方法解决数学问题。)
Source: Oxford Engpsh Dictionary, Merriam-Webster Dictionary.