It seems the confusion is betweenno happier andnot any happier . The difference becomes apparent if we insert the wordmore , which is implicit. The second form is actuallynot any more happier which conveys the same meaning asno happier . As happier is the comparative form of happy, so we don't need a second comparativeany more .
For eg.
X is not taller than Y. - concise
X is not any (more) taller than Y. - too verbose
...
For eg.
X is not taller than Y. - concise
X is not any (more) taller than Y. - too verbose
...
