Hello everyone. I am writing this after much thinking.
As this two words are similar but having some difference. Like in INDIA, or particularly say in HINDI language belief is called-"Shraddha" and trust is called -"Viswas". So, often people use words that-"Whatever religion you follow, your SHRADDHA in your God should be there. Then and then only religions are important. If SHRADDHA is not there then nothing you will get.
However word SHRADDHA is something different than BELIEF but I am here taking them same to explain the difference. So here on the top as I mentioned, most of I heard of BELIEF IN GOD than TRUST in GOD.
SO difference are that as I listed some below.
Belief is basically connected to the heart. Whether there is legacy or myth about anything, whether you have experienced it or not, but you believe in that. Like, for example, I just by my intution believing that Germany should be best. I like Germany. I had never been in the past there. yet my belief was strong in that. Later it came true and I came to know that Germany is best in mechanical engineering. Most of intelligent scientists are from there only in mechanical side. And yesterday even BAYREN MUNICH -football team won against BARCELONA. I never played nor watched the game of football, yet I had belief in evruthing of Germany. So, that's what all about BELIEF. You haven't experienced, yet you believe.
Trust is related to mind first than heart. Means, first you want experience and then you trust it. Like you don't trust everyone you haven't seen and haven't met. Even after single meeting you don't get fully trust. Trust is something called-"TRIED AND TESTED AND EXPERIENCED BELIEF".Like a particular man can cook the best recipie, that you only can believe once you test it. So, later even someone challenges you, you can firmly took stand and say to him boldly that, no I have trust on him, HE can cook. So, you can't sell TRUST without testing. If you're giving, it is called your SHRADDHA in that person(here BELIEF cant be used). So,TALKING MORE IN TERMS OF HUMAN, TRUSTING IS THE BELIEVING OF ONES ATTITUDE, CHARACTERISTICS, QUALITY, BEHAVIOUR and all that. While in BELIEF-SHRADDHA, THERE IS NO QUESTION OF QUESTIONING THIS TERMS OF HUMAN. EVEN for non-living things, TRUST is about questioning their quality and all that included in process of TRUST. I think, no further explanation is needed to explain trust. Because, you are well known familiar with TRUST.
So, according to me what you get by your intuition is right because it is your inner voice. Whatever you trust is like -you have questioned that truth or the phenomena. So, you never ask for proof of your general late coming of your partner. You never question your GOD to do magic for you. Isn't it? You believe them without any proof. That's important. That's shows your faith in that person. And most of relations are formed here on that basis only.(I know about INDIA, rest I don't know much, but mostly it would be same).
So according to me, though you look SHRADDHA without proof as a weak, but it shows your pure faith. Perhaps you have heard this words-"If you do questioning/judging the people, you don't have time to love them." So, real or faithfulness grows in SHRADDHA not in TRUST.
This are my personal views. May be your views are different. You can tell me yours...
Have a great day :)