Why is the spelling of "pronounce" and "pronunciation" different?

pro-NOUN-ciation is universally wrong. Even the highly permissive Merriam-Webster dictionary marks it with an obelus (÷). Here is what they say about pronunciations marked with an obelus:

The obelus, or division sign, is placed before a pronunciation variant that occurs in educated speech but that is considered by some to be questionable or unacceptable. This symbol is used sparingly and primarily for variants that have been objected to over a period of time in print by commentators on usage, in schools by teachers, or in correspondence that has come to the Merriam-Webster editorial department. In most cases the objection is based on orthographic or etymological arguments. (source)

As for why the word pronounce has an O between the two N’s and pronunciation does not, it is unclear, but both words derive from French, pronunciation from pronunciation and pronounce from pronuncier. There is probably some variation in the way the different word stress affected how the words were spelled after being borrowed into English.

Trisyllabic laxing is the reason for the pronunciation difference, which led to the spelling difference.

English words descended from Latin "nuntiare ('to announce')" are non-uniform. I've bolded the irregularities:

pronounce | pronunciation | pronounciation | pronouncement | x

announce | annunciation (archaic) | x | announcement |x

denounce | denunciation | x | x | x

renounce | renunciation | x | x | x

X | enunciation | x | x | enunciate

It is scarcely more defensible to say that "pronounciation" is wrong than to say that "enunciate" should be replaced by "enounce" or "announcement" by the more venerable "annunciation".

Contra the top answer, pro-NOUN-ciation is a fine pron(o)unciation, as it is common and easily-understood. It may be regretted by sticklers, but this is the way millions of native speakers say it.

The spelling "pronounciation", however, faces intractable stigma (spelling variants are generally tolerated less than pronunciation variants). It is best avoided in formal writing.