We often use these phrases interchangeably.
But they are not the same.
"I believe" can also mean "I guess." "I don't know for sure, Dude, but I believe ... " yada and yada and ya da. Knowledge invokes certainty, confidence.
But knowledge doesn't necessarily imply relational trust.
"I know" can also mean "I know, but I'm not going to do anything about it" or "I know I should trust you, but I just can't find it in my heart to believe you."
When Martha, grieving the loss of her brother, said "I know he will rise again in the resurrection at the last day," I find myself wondering about the state of her belief.
And Jesus does press her. He doesn't leave her in her knowing, but pushes her for that sort of belief that involves trust and maybe even faith.
What do you think? Can you interchange "I know" and "I believe"?