Difference between "believe" 认为 and 相信?

认为 comes from thought, inference, reasoning (and sometimes beliefs or trust also).

One says 认为 when he usually has some reason/inference to support his opinion.

相信 only comes from beliefs or trust.

相信 means one trusts something and has no reason/inference but just beliefs or trust.

BTW, 觉得: one has no belief in his opinion and also has no confidence in his reason/inference. That's feel like.

认为 like think

相信 like believe

觉得 like feel like

想 could be think/believe/guess/want/... (depended on context), 想 is really wide in usage

我想去吃晚饭 I want to dinner

我想他是对的 I think/believe/guess that he is right (depended on context)

我认为他是对的 I think that he is right (I have reasons/inference)

我相信他 I trust him

我相信他是对的 I believe that he is right (I just trust it and I have no inference)

You cannot say 我認為你 however 我相信你(I believe in you) is acceptable. If you want to use 認為, 我認為你是對的(I think you are right.) is acceptable.

And in Taiwan, According to dictionary owned by Ministry of Education.

認為 means:

對某一事物經分析思考後所作的判斷。

(Judgement to something after analysis and thought.)

相信 means:

信任，以為可信賴。

(Trust, thinking (something) is trustworthy.)

以為確實如此而不懷疑。

(Thinking something is the case and no doubt.)

The key character in 相信 is 信 which means faith/trust. (相 is just there to turn 信 into a standard two-character word. It can be omitted in spoken language.) I.e. there's some emotion involved. The object can also be a person or a belief (i.e. equivalent to "believe in").

For example, 我相信你 means "I trust you"/"I have faith in you/what you said".

认为... is literally "recognise ... as" and can be roughly translated as "have the opinion that". It involves no emotion, but a mere statement of thought. Also, it must be followed by a statement clause and not a simple subject.

Compare and contrast

相信 and 认为 are not interchangeable due to context:

• 我相信未来会更好 （Using 认为 will sound presumptuous)
• 科学家认为大爆炸是宇宙的起源 (相信 will make the belief sounds less scientific and more wishful thinking)

• 我相信我儿子是不会做坏事的
• 警方认为你儿子触犯了法律 (认为 is treated as stating a fact. When used with a character judgement, some kind of authority or evidence is required. Otherwise, it will sound libellous.)

They are not interchangeable due to grammatical structure:

• 我相信进化论 (= I believe in evolution.)
• 我认为进化论是真理 (= I believe that evolution is true.)