"I don't believe this" expresses no doubt. The speaker simply doesn't believe it. "I can't believe that" leaves a little room for doubt. The speaker may have a little doubt. This is also used to express astonishment. For example, "I can't believe she died!" doesn't mean that the speaker doesn't believe it. The speaker knows it's true, but is astonished or shocked. Hope this helps Anna.

There is very little difference between the two statements. To hair split, one can say 'I don't believe this' is an absolute rejection while 'I can't believe this' is less absolute and has a connotation of disbelief in something one has read or seen. NB The intonation and stress when spoken is important and gives a guide to what the speaker means.

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