Not an historian, but...

It would differ wildly to the point of being a completely different language, since in Rome at the time of Caesar (any of them really) they would speak ancient Latin and in the case of the high class, they also spoke ancient Greek. These languages however bear little resemblance with modern Italian or modern Greek, having suffered huge transformations in the course of two millennia.

Latin as it turns out gave rise to such diverse languages as modern Italian, but also French, Spanish, Portuguese and Romanian just to name a few. Could a fluent Romanian speaker just show up in Paris and be understood (in French that is)? Not at all. Ancient Latin is at least as diverse from modern Italian as modern Italian is from modern Romanian. (This is an approximation, a linguist will hopefully shed some more light on it)

For a comparison I would image you are fluent in some form of Modern English. Then try to read some Old English: Not from 20 centuries ago, but just a mere 10 centuries ago: In this case Beowulf. I would say that you can understand about same of that text as a modern Italian would of ancient Latin.

Edit: just for comparison a Poem from Catullus in both Latin and modern Italian:

Latin:

Cui dono lepidum novum libellum

arida modo pumice expolitum?

Corneli, tibi: namque tu solebas

meas esse aliquid putare nugas.

Iam tum, cum ausus es unus Italorum

omne aevum tribus explicare cartis...

Doctis, Iuppiter, et laboriosis!

Quare habe tibi quidquid hoc libelli—

qualecumque, quod, o patrona virgo,

plus uno maneat perenne saeclo!

Italian:

A chi dono il mio libretto nuovo (ed) elegante

appena levigato dalla dura pietra pomice?

A te, Cornelio: e infatti tu eri solito

pensare che le mie sciocchezze valessero qualcosa

già allora, quando hai avuto il coraggio, unico fra gli Italici,

di esporre tutta la storia in tre libri

eruditi, per Giove, e laboriosi.

Per questa ragione prendi questo libretto, qualunque cosa

e quale il suo valore (sia); perchè nella vergine madre

possa rimanere più di un solo secolo.

Latin Language and Classics scholar here. I speak Latin and converse in it regularly with other similarly pretentious scholars and students from various places in Europe.

From my experience, you would likely not be very well understood. They would definitely be able to hear similarities and perhaps even recognise a few words here and there, but if you were just talking at them in Italian, they would probably not comprehend much, if anything, of what you were saying. The Latin accent was different to the Italian accent and Italian has other influences and dialects mixed in that Latin does not share. Also, a lot of time has passed since Latin was used and language changes quite quickly (as you can see from the differences in English between now and the 1800's, for example).

As someone who can speak Latin, I can understand only rudiments of Italian when I hear it spoken. If I read it, I can understand a little more, but it is by no means the same language.

The differences between Latin and Italian are far greater than those between American modern English and 1800's Queen's English. It's more akin to the difference between Italian and Spanish, or Italian and Romanian - these languages have a lot in common, and if you speak one you may be able to pick up slivers of meaning from the others, but they are, at the end of the day, two different languages.

On the other hand (to flip this question around) I have on occasion spoken Latin to modern day Italians (I thought it would be interesting to see how much they understood of it). Again, it sounds kind of familiar to them, but not to the extent that they can follow a conversation or understand what I might be talking about. Just bits and pieces, a few words here and there are identical, others are merely similar, many are totally unfamiliar. The grammar is quite similar, but that is not necessarily something you would pick up on if you are hearing it only conversationally.

I hope this helps answer your question!