Difference between "believe" 认为 and 相信?
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 &quot;believe in&quot;).
For example, 我相信你 means &quot;I trust you&quot;/&quot;I have faith in you/what you said&quot;.
认为... is literally &quot;recognise ... as&quot; and can be roughly translated as &quot;have the opinion that&quot;. 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.)