Historically speaking, both Sicily and Italy are a melting pot of sorts. Early European history tells us that the Italian peninsula has been settled, for varying periods of time, by a large number of Europe's early, pre national, groups. Most streaming in from Eastern, Northern and Western Europe.
Sicily, like Italy, because of it's strategic geographical location, had also been over-run by various tribes, and pre national groups. The old saying is, if Italy has been conquered every 100 years, then Sicily has been conquered every 50.
In the golden age of trade by sail, Sicily was the gateway to Europe. Sicily was to the shipping lanes what Persia was to the Silk Road.
The major difference is this. Where Italy was conquered often by the Western, Eastern and Northern peoples of Early Europe, Sicily was conquered by the people of the Byzantine (Turkish/Greek), Greek, and Saracen (an early Arabic people often confused with the Moors).
For most of it's history Sicily has been under Greek influence and during the Golden Age of Greece, Sicily was part of Greece. Archimedes and many other famous Greek philosophers were born and lived on the island of in Sicily.
Because of this Sicily has two major roots of both culture and blood, that being Greek and Arabic.
The major exception to the Sicilian puzzle is the tremendous influence of the later conquering Norsemen (Vikings) which sacked Sicily. As a matter of fact, though Sicily appreciated a kind of autonomy for a good portion of it's history, though under the rule of varying peoples, the first King of Sicily was actually a Norseman.
This is sometimes confused by the fact that these new invaders were known as Norman from Normandy, but this was so early in Norman history that most of these people actually were still of a Viking heritage, of Danish stock or merely first generation Norman. This being important from cultural perspective because they brought to Sicily not the Frankish culture which was quickly adopted by the new Normans but rather their traditional Scandinavia culture. This being the only instance of a relevant northern European culture.
Even the languages of Sicily and Italy are completely different. Sicilian is not simply a dialect of Italian, as is often stated. The Sicilian language is linguistically older and of a different root than Italian. Sicilian was a recognized language while the literate people of the Italic peninsula were still speaking Latin, while the masses spoke regional languages (later morphing into the many dialects we see there today). The Italian language of today, with the majority of it's dialects, was derived from the Latin base combining the most influential languages brought to the peninsula.
Sicilian,in comparison shows a stronger connection to the earlier root of Indo-European spoken by middle eastern, Greek, and Arab peoples.
It was only very recently, in 1870 that Giuseppe Garibaldi finally managed to unify the Italian peninsula, and with it the islands of Sicily, into one nation.
Prior to Garibaldi, Italy was a collection of nation-states and historically wared over parts of the Southern Italian Peninsula with Sicily. At it's autonomous height as a nation, the Italian peninsula from Rome down an Sicily made up the Kingdom of Two Sicilies.
Today Sicily and Italy are one nation, Italy. The major differences in cultural background is what still confuses some people and why the Sicilian heritage is still strongly seen as different from the Italians of the Peninsula.