The two words nature and natural have the same root, so why are they pronounced differently?
As Greg Lee and Peter Shor mention in the comments, the "shortness" of the first vowel in natural /ˈnætʃ(ə)rəl/ can be explained by the concept of "trisyllabic laxing": a stressed vowel in the third-to-last syllable of a Latinate word is often given its short pronunciation even if it is only followed by a single consonant. This also affects the pronunciation of the adjectives national (related to the noun nation) and rational (related to the noun ration).
The pronunciation of nature as /ˈneɪtʃə(r)/ with a long vowel is actually irregular; most of the time, vowels in Latinate words are shortened before a "u" in an (orthographically) open syllable, as Peter Shor suggests in another comment (consider tenure, figure, stature, spicule, ferule). I don't know if there are any details about the history of this word that would explain why it does not rhyme with stature or mature.
Nature has the silent e at the end, and thus the a is pronounced long. (a long pronunciation of a vowel is one that sounds like the name of the vowel, e.g. long a is pronounced ay, long e is pronounced ee, etc.) A few examples of this are cove (o is long), calculate (u and second a are long), and squeeze (the u and the double e are long).
Natural does not have a silent e at the end, and thus, the first a is not pronounced long. Now, however, if one was to add a silent e at the end, the last syllable would be pronounced as one would pronounce "ale", in which the a is pronounced long.
source: my education, http://grammar.yourdictionary.com/word-lists/english-words-end-with-silent-e.html for the example words