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I know both 觉得 and 认为 can be used to mean "to think", like:
你认为这篇文章怎么样？
你觉得这篇文章怎么样？
But most people would translate 觉得 to "to feel". Does 觉得 express more of what someone thinks based on just feeling, are there any cases where these two words are not interchangeable?
Yes, 觉得 expresses more of what someone thinks based on feeling. 认为 expresses more of a person's opinion based on considerations and beliefs. As such, 认为 is always stronger than 觉得 when communicating thoughts. There are cases where the two words are not interchangeable:
When expressing a strong belief:
在过去，人们认为喜欢抛头露面的妇女是不守妇道的。
In the past, people considered women who appear in public to be lacking in virtue.
When expressing a feeling:
我觉得很遗憾，这件事不该发生的。
I feel very sorry. This incident should not have happened.
Where they are meant to communicate thoughts, although interchangeable, there is always a preference of one over the other:
When expressing a casual thought without a line of reasoning:
我觉得这篇文章太深奥了，就算让我再读几遍我也读不懂。
I feel that this article is too profound. I wouldn't be able to comprehend even if I read it a dozen times.
When expressing an opinion with a clear line of reasoning:
我认为这篇文章太深奥了，使用的词汇不是一般人都能看懂的。
I opine that this article is too profound. The vocabulary used is not what an ordinary person can understand.
我觉得很累. I feel so tired.