Trust/Believe : “信任” vs “相信”

While 信任 can only be used for a person (or a group of people), 相信 can be used for both a person or a non-person object. I will here discuss what's the difference when used for a person.

信任 focuses on describing the characteristic of the person. When you say "我信任小张", that means 小张 has your endorsement to be a trustworthy/capable person.

When you say 我相信小张, it means you believe in 小张. It focuses on your action of believing instead of on the personality/ability of 小张.

You may tend to use 信任 over 相信 if your relationship with 小张 is close. However, even with this said, the two sentences above are describing almost the same situation.

「相信」这一词指的是对某件具体的事情，「信任」指的是对某一个人

The book 1700对近义词语用法对比 (item 1358, pages 1256 and 1257) compares these two words. They write:

“信任”和“相信”都是动词。“信任”的对象是人（或单位、组织等由人组成的团体），信任某人，表示认为他人品好、有办事能力，因而让他去负责某事；“相信”的对象可以是人，也可以是事，相信某人，表示对某人了解，对他说的话或做的事不怀疑，相信某事，即认为它是真实的。

Here they emphasize that 信任 only applies to other people (or organizations etc. created by people), believing that person's character is good, whereas 相信 (unlike 信任) can apply to people's speech, and other matters that you trust are true.

Their example sentences:

1. 公司领导很 [相信 / 信任] 他，最近让他当了一个部门的经理。
2. 既然用他就应该 [相信 / 信任] 他，用人不疑嘛。
3. 我不 [相信] 他说的话。
4. 你这么努力，我 [相信] 你一定能学好汉语。
5. 我 [相信] 这件事是真的。

They also highlight how the object of 信任 cannot be the subject (unlike 相信). More specifically they write:

“相信”的对象可以是自己，“信任”的对象不能是自己。
相信's object can be oneself, 信任's object cannot be oneself. [my translation]

giving the example

我 [相信] 自己的判断是正确的。

I'm not a native Chinese speaker but I've always used 相信 to mean believe (related to truth/falsity) and 信任 to mean trust/confidence (related to success/failure).

我相信你 I believe you. I believe what you just said. I think you are telling the truth. I don't think you are lying.

我信任你 I trust you. I trust in your ability. I have faith in you. You won't let me or yourself down. I am confident that you will succeed. I am confident that you will not fail.