First of all, this is not true exclusively of Italian cities. Other citiies like Copenhagen, Prague, Moscow, and Belfast also have names significantly different in their source languages (København, Praha, Moskva, Béal Feirste).

Why? Four reasons. The first is normal sound changes in English. All words in the language are subjected to these, even the names of cities, and the longer the city has had a name in English, the more sound changes it would have gone through. So an old famous city is more likely to have experienced more sound changes. Even English cities are subject to this - Manchester was Mameceastre in Old English, for example.

The second is sound changes in the foreign language that occurred after the name was established in English. This is a particularlarly large factor in Italian cities. Florence, for example, used to be Florentia in Latin. English has actually preserved a much closer version of that than Italian has. And why is Livorno called Leghorn? Because only a few centuries ago it was called Legorno in Italian. (And actually, the discrepancy with the Italian name is getting too large, and Leghorn is not famous enough for its English name to have much inertia, so the use of the name Leghorn is dropping off in English as the word is reimported as Livorno.)

The third is sound changes in intermediate languages. English borrowed a lot of words from French, which went through very extreme sound changes from the original Latin over the last 1500 years. Not to get too far off track, but that's why we often have two versions of some Latin words in English - one borrowed directly from Latin (e.g. masculine) and one borrowed from French after that same word has been put through the meat grinder of French sound changes (e.g. male). So some Latin cities like Roma were actually changed in French, e.g. to Rome, and then borrowed to English. This is also true of Venice, where the change from Venetia to Venice happened in French, while a different change in Italian led to the modern Venezia. (In the case of Copenhagen, we borrowed the German name for the city rather than the Danish one.)

Lastly, there is also simple Anglicization, in which the city name is altered to make it simpler for people accustomed to English phonology to pronounce. This is the case in many Irish and Scottish cities, but also true to a lesser degree in Italy. In Italy it was normally a case of lopping off the vowel at the end, as Italian phonology is in general not difficult for English speakers to approximate, except for maybe the initial "gl" sound. So Latin Syracusae became English Syracuse (and Siracusa in Italian).

I can maybe help with a perspective from the other side. In Italy a lot of city names from UK, Germany and France are Italianised: London/Londra, Berlin/Berlino, Paris/Parigi. It was a typical thing until at least the 1500s so cities that were part of everyday conversation were 'adapted' to sound less foreign. Spanish cities were asked too but that's less obvious as the pronunciation is already pretty similar Barçelona/Barcellona, Sevilla/Siviglia. On the other hand cities that entered common lexicon later on have more or less retained their original pronunciation: New York, Sydney, Tokyo

There are exceptions but that's a more or less general rule which I think could work in English as well