I think there is almost no difference.
One difference I can think of is, when the proposition is positive (for example, "I just won the lottery"), "I can't believe" sounds more natural. When it's negative, there's no difference.
Examples:
"I can't believe I lost my keys again" O
"I don't believe I lost my keys again" O
"I can't believe I got the job" O
"I don't believe I got the job" X