In English, you can drop a number of words, but in French you cannot do it.
So if "I believe it is true" is a shorter version of "I believe that it is true", there is no correct shorter version of the French sentence.
the structure works as follows:
main clause: je crois
subordinating conjuction: que
subordinate clause: c'est vrai
While they do have separate definitions (see other response), they are near synonyms and are interchangeable most of the time, as they are in English. There are some occasions where only croire should be used, some where only penser should be used, but they would correspond with English.