Difference between "This makes no sense" and "This doesn't make sense"

Solution 1:

At a first approximation they mean the same thing, but there is an important difference.

"This doesn't make sense" is simply pointing out an inconsistency in whatever is being discussed. Eg, perhaps a column of numbers does not add correctly. Or, alternatively, it could be pointing out that phrase or sentence is somehow garbled and hard to understand.

"This makes no sense" is saying that substantial parts of the materials being discussed are inconsistent or impossible to understand. In essence it's saying "This is nonsense".

Solution 2:

These phrases are equivalent in meaning and a google ngram query indicates roughly the same frequency of usage.

Oldcat's comment does reveal a subtle distinction if you require emphasis.