Ambiguous meaning? (Logical thinking required)

We probably are all aware of the difference between saying "I don't believe in the existence of God", and "God does not exist". One is a statement showing uncertainty, and the other is a statement expressing certainty.

 Someone said this : "Do people understand the difference between saying I don't think what you're saying is true and I think what you are saying is false?"

Is there a difference in meaning between :

" I don't think what you're saying is true" and
"I think what you are saying is false"?

Of course, people often just mutter words that don't often reflect what is meant. Careful orators or writers do their best, but this is the exception, not the norm. This is why clarification is so important. "What do you mean by that?" And trying to "steelman" their argument or statement.

Let's break this down:

I don't think...

Expresses uncertainty as well as the fact that one is making an epistemological claim, not an ontological one.

...what you're saying is true.

The options are: the speaker either a) thinks what is being said is false or b) does not think it is true or false, but does not know. Again, people rarely say what they mean. I would estimate in almost all cases, the speaker actually means that what is being said is false.

So is there is a difference? Yes... in a strict semantical sense. In practicality, no difference.

If I say, "I am certain I do not believe in the existence of god," does that change your question?